UAV Flight Path Planning Based on HPSOCAOA Optimization Algorithm

Abstract

To address the issues with the Crocodile Ambush Optimization Algorithm (CAOA) in UAV trajectory planning—such as its tendency to get stuck in local optima, the difficulty in balancing global search and local exploration, and low convergence accuracy—this study proposes a three-dimensional trajectory planning method based on the Hybrid Particle Swarm and Crocodile Ambush Optimization Algorithm (HPSOCAOA). First, a collaborative search structure combining the Particle Swarm Optimization (PSO) algorithm and the Crocodile Ambush Optimization Algorithm (CAOA) is established; second, an adaptive energy consumption coefficient is designed to address the issues of premature individual elimination in the early stages and insufficient convergence momentum in the later stages, thereby further balancing global exploration and local exploitation; finally, crossover learning is introduced. Using a cross-group replacement mechanism for superior individuals, PSO’s fine-tuning identifies high-quality individuals, which are then substituted for lower-quality individuals in CAOA. This resolves the problems of redundant low-quality individuals within the population and low search efficiency, and enhances overall optimization performance. Standard test functions demonstrate that HPSOCAOA outperforms the comparison algorithms in terms of optimization accuracy and stability. In simulation experiments for path planning in complex 3D mountainous environments, HPSOCAOA was compared with classical intelligent algorithms, verifying its superiority and practicality in complex 3D scenarios.

1. Introduction

3D flight path planning for drones is essential to the safe and efficient operation of low-altitude aircraft and underpins the rapid growth of the low-altitude economy [1]. With the gradual opening of low-altitude airspace, typical scenarios including urban logistics [2], emergency communications [3], and aerial inspection [4] have raised higher requirements for UAVs in operational density, environmental adaptability and real-time planning capability [5]. In such complex operating environments, designing feasible, safe, and optimal flight trajectories has gradually become a research hotspot in the field of autonomous flight.

Traditional path planning methods are mainly based on graph search and random sampling techniques, such as A*, RRT, and RRT* algorithms [6]. These methods have clear logic and high interpretability, but they are prone to node redundancy, low planning efficiency, and difficulty obtaining globally optimal paths when dealing with high-dimensional complex scenes. Therefore, they are increasingly difficult to meet the real-time and optimality requirements of modern UAV systems.

In recent years, swarm intelligence optimization algorithms have been widely used in UAV path planning due to their strong global search ability, fast convergence speed, and no dependence on gradient information. Among classical metaheuristic algorithms, the Ant Colony Optimization (ACO) algorithm excels in global search capability but converges slowly [7]. To overcome the limitations of classical algorithms, a variety of improved strategies have been proposed and verified.

An adaptive hybrid particle swarm optimization (PSO) algorithm was proposed in [8], which demonstrated outstanding performance in mountain peak path planning models. Reference [9] integrates the sine–cosine function and Lévy flight strategy. By dynamically adjusting the search step size and occasionally introducing large-step jumps, it enhances the symmetry balance of the algorithm and effectively improves the global optimization performance. Reference [10] proposes a hybrid UAV path planning framework named PSO-RRT, which integrates the PSO guidance mechanism and introduces the time-varying acceleration coefficient and path sparsification strategy. It effectively improves path quality, shortens planning time, and reduces computational cost, and is suitable for real-time UAV path planning in complex constrained environments.

Reference [11] adopts a multi-strategy improved beetle algorithm, significantly enhancing the algorithm’s convergence speed and optimization capability. An enhanced multi-mechanism beetle algorithm was proposed in [12], which integrates chaotic mapping, dynamic global search, and adaptive and dynamic weighting mechanisms to improve convergence speed and accuracy. Reference [13] achieves an adaptive adjustment between global exploration and local exploitation by dynamically balancing the gray wolf operator with the genetic operator, enabling the algorithm to demonstrate higher optimization efficiency, stability, and robustness.

The improved Dubins-CPSO algorithm proposed in [14] retains the high reliability and low mutation rate of traditional algorithms while maintaining the flexibility and robustness of intelligent algorithms, thereby achieving complementary advantages. The adaptive elite binary particle swarm algorithm proposed in [15], which employs continuous convex approximation and Q-learning, outperforms various other intelligent algorithms in terms of joint mission performance. References [16,17] proposed an improved whale optimization algorithm and developed a two-layer trajectory planning method suitable for multi-UAV delivery on this basis. A series of swarm intelligence algorithms with different improvement strategies have yielded fruitful results in the field of UAV trajectory planning [18,19,20,21,22], but most of them still face limitations in optimization accuracy, convergence speed, and stability when dealing with complex multimodal problems In contrast, swarm intelligence optimization algorithms have attracted widespread attention due to their strong robustness and efficient global searching ability. Although a large number of improved intelligent strategies have been proposed, most existing algorithms still suffer from inherent limitations. In 2025, a novel metaheuristic algorithm named Crocodile Ambush Optimization Algorithm (CAOA) was proposed, which has shown competitive performance on benchmark test suites but still has obvious defects and has not been applied to UAV 3D path planning [21].

To address the above shortcomings, this paper develops a hybrid optimization algorithm named the Hybrid Particle Swarm and Crocodile Ambush Optimization Algorithm (HPSOCAOA). The proposed framework integrates the search mechanisms of CAOA and PSO, designs an adaptive energy-consumption coefficient, and a cross-learning elite replacement strategy to enhance optimization performance. Comparative experiments demonstrate that the presented HPSOCAOA achieves significant improvements in convergence accuracy, computational stability, and optimization efficiency, which verifies its theoretical superiority and practical engineering application value.

2. Related Work

In the field of algorithm optimization for UAV path planning, our research group has conducted a series of relevant explorations. Reference [9] integrates the sine–cosine function and Lévy flight strategy, enhancing the algorithm’s symmetry balance by dynamically adjusting the search step size and occasionally introducing large-step jumps, thereby effectively improving global optimization performance. Building on this work, references [16,17] propose an improved whale optimization algorithm and construct a two-layer trajectory planning method suitable for multi-UAV delivery tasks. Furthermore, reference [23] extends our research to three-dimensional scenarios, carrying out UAV 3D trajectory planning based on the starfish optimization algorithm and providing a new solution framework for path optimization problems in complex environments.

In 2025, Xu et al. [24] proposed the Crocodile Ambush Optimization Algorithm (CAOA), inspired by the ambush behavior of crocodiles. This algorithm has shown competitive performance on the CEC2017 test function, but it still suffers from deficiencies, including insufficient initial population diversity, poor balance between global search and local exploration, a tendency to get stuck in local optima, and limited convergence accuracy. Furthermore, CAOA has not yet been applied to UAV trajectory planning, which limits its practical application value.

Therefore, it is of great theoretical significance and practical value to improve CAOA and extend it to solve UAV 3D flight-path planning problems, which is also the core research content of this paper.

3. Modeling

3.1. Environmental Modeling

The environmental model area for path planning is set to a Cartesian coordinate system. In this paper, the 3D coordinates are planned as 1000 × 1000 × 1000. To better reflect real-world scenarios, a mountain model is adopted as the obstacle, and its mathematical expression is given as follows:

$$Z(X,Y)={\displaystyle {\displaystyle \sum _{i=1}^n}{h}_i\exp \left[-{\left(\frac{X-X_i}{X_{si}}\right)}^2-{\left(\frac{Y-Y_i}{Y_{si}}\right)}^2\right]}$$

(1)

In the equation: $\left(X_i,Y_i\right)$ is the center coordinate of the $i$-th peak; $h_i$ is the terrain elevation parameter; $X_{si}$ and $Y_{si}$ are the attenuation and control slope of the $i$-th peak along the $x$ and $y$ axes, $n$ respectively, represents the total number of peaks.

In mountainous and urban environments, typical obstacles such as transmission towers, no-fly zones, and cylindrical barriers can be reasonably abstracted as vertical cylinders. Cylinder modeling is a standard and widely recognized method in the field of UAV path planning. This modeling approach can enclose actual obstacles while reserving sufficient safety margins, which is suitable for simulation verification and possesses typical representativeness. To simulate a realistic, complex mountainous environment, mountain surfaces and cylinders were constructed within the coordinate system to represent the terrain and obstacles in the flight environment, as shown in Figure 1.

Figure 1. Schematic Diagram of the Environmental Model. The pink area represents the no-fly zone.

A total of 5 cylinders were set up; the obstacle information is shown in Table 1.

Table 1. Obstacle Information Table.

Obstacle Number	Center Coordinates	Height (m)	Radius/Dimensions (m)	Type
1	(600, 100)	150	40	cylinder
2	(500, 350)	150	50	cylinder
3	(750, 350)	150	70	cylinder
4	(150, 200)	150	40	cylinder
5	(450, 550)	150	20	cylinder

3.2. Cost Model

Under normal circumstances, drones have a defined safety distance. The critical safety zone of a drone is modeled as a sphere with a radius of $d_1$, while the danger zone is modeled as a sphere with a radius of $d_2$. Set the safe distance for drones $d_2=3d_1$. As shown in Figure 2, when the distance between an obstacle and the drone is greater than $d_2$, there is no risk of collision; when the distance between the obstacle and the drone is greater than $d_1$ less than $d_2$, there is a risk of collision, and when the distance between the obstacle and the drone is less than $d_1$, a collision has already occurred.

Figure 2. Schematic diagram of the drone model.

3.2.1. Path Length Cost $V_c$

The length of a drone’s flight path is often a decisive factor in flight efficiency and value; therefore, we model flight length. The model specifies that each drone has a maximum allowed length, and if the planned total path length approaches this maximum, the cost associated with length increases.

Considering the total flight distance from the starting point to the endpoint as the sum of individual arc segments, and assuming the total flight path is the sum of n sub-segments, the expression is as follows:

$$\begin{array}{lll}{V}_c& =& {\displaystyle \sum _{i=1}^{n-1}{L}_i},\\ L_i& =& \sqrt{{\left(x_{i+1}-x_i\right)}^2+{\left(y_{i+1}-y_i\right)}^2+{\left(z_{i+1}-z_i\right)}^2}\end{array}$$

(2)

A drone waypoint is represented as a set of discrete waypoints $\left(x_i,y_i,z_i\right)$, $L_i$ represents the distance between the drone’s $i$-th waypoint and $\left(i+1\right)$-th waypoint.

3.2.2. Terrain and Obstacle Costs $V_s$

To ensure that the drone’s flight path avoids obstacles, this is achieved by adjusting $V_s$ the control value. The obstacle area is modeled. The degree to which an obstacle threatens flight safety is related to its distance. Obstacles are simulated as cylinders of varying heights. The center coordinates of an obstacle are set to $\left(x_o,y_o\right)$. For a waypoint coordinate $\left(x_i,y_i\right)$, we define $p_i$ as the distance from the waypoint coordinate to the center coordinate of the obstacle.

$$p_i=\sqrt{{\left(x_i-x_o\right)}^2+{\left(y_i-y_o\right)}^2}$$

(3)

$$r_{collision}=r+d_1$$

(4)

$$r_{threat}=r+d_2$$

(5)

$$R_{ij}=\left\{\begin{array}{cc}0& p_i>r_{threat}\\ r_{collision}+d_1-p_i& r_{collision}<p_i<r_{threat}\\ 1000& p_i<r_{collision}\end{array}\right.$$

(6)

$$V_s={\displaystyle {\displaystyle \sum }_{i=1}^N}{\displaystyle {\displaystyle \sum }_{j=1}^T}{R}_{ij}$$

(7)

The $r$ is the radius of the obstacle, the $r_{threat}$ represents the threat distance, which is used to assess the level of risk between the drone and an obstacle; the $r_{collision}$ is used to determine whether a collision will occur between the drone and an obstacle; the $V_s$, $R_{ij}$ represents the constraint from the $i$-th waypoint to the $j$-th no-fly zone. A severe penalty is imposed when $p_i$ is less than $r_{collision}$.

3.2.3. The Cost of Flying High $V_h$

Many factors must be considered when determining a drone’s flight altitude. To ensure stealth, image clarity, and low energy consumption, it is desirable to maintain the lowest possible altitude; however, for safety reasons, the drone’s flight altitude must not fall below the safe range. The flight altitude problem is modeled as follows.

$$V_h={\displaystyle \sum _{i=1}^{N-1}\left\{\begin{array}{cc}\left|h_i-\frac{\left(h_{\max }+h_{\min }\right)}{2}\right|& h_{\max }\le h_i\le h_{\min }\\ 1000& \mathrm{others}\end{array}\right.}$$

(8)

In the equation: $h_{\max }$, $h_{\min }$ represent the maximum and minimum flight altitudes, $h_i$ is the flight altitude of the drone at the $i$-th waypoint. An excessive penalty is imposed once the flight height exceeds the preset range, as shown in Equation (8).

3.2.4. Cost of Pitch and Yaw Constraints $V_f$

The pitch angle is defined as the angle between a vector parallel to the aircraft’s axis and pointing in the direction of flight, and the ground. During path planning, the drone’s ascent or descent must not be too steep; therefore, it is necessary to model the pitch angle of the flight path.

$$v{1}_i=P_{i+1}-P_i$$

(9)

$$v{2}_i=P_{i+2}-P_{i+1}$$

(10)

$${\delta }_i={\cos }^{-1}\left({\displaystyle \frac{v{1}_i·v{2}_i}{‖v{1}_i‖‖v{2}_i‖}}\right)$$

(11)

$${\gamma }_i={\tan }^{-1}\left({\displaystyle \frac{z_{i+1}-z_i}{\sqrt{{\left(x_{i+1}-x_i\right)}^2+{\left(y_{i+1}-y_i\right)}^2}}}\right)$$

(12)

$$V_f={\displaystyle {\displaystyle \sum }_{i=2}^N}{\omega }_{\delta }·{\delta }_i+{\displaystyle {\displaystyle \sum }_{i=1}^{N-1}}{\omega }_{\gamma }·{\gamma }_i$$

(13)

where $V_f$ represents the magnitude of the angle of change in all directions, used to measure the smoothness of the path; $P_n$ represents a waypoint; $v{1}_i$ and $v{2}_i$ represent the vector difference between two adjacent points, with the direction of the vector being the same as the UAV’s flight direction; ${\delta }_i$ denotes the inverse cosine of the angle between $v{1}_i$ and $v{2}_i$, i.e., the turn angle; ${\gamma }_i$ denotes the pitch angle; ${\omega }_{\delta }$ and ${\omega }_{\gamma }$ denote the weights of the turn angle and pitch angle, respectively. Based on Equations (9)–(13), the pitch angle of the UAV can be calculated. When the difference in pitch angle between adjacent flight segments exceeds the threshold of 45°, the excess angular deviation is taken as the penalty value. A larger angular deviation corresponds to a greater smoothness penalty.

3.2.5. Spatial and Boundary Costs $V_b$

To ensure that drones do not collide with other drones during flight, a penalty is applied when the spatial distance between drones falls below a safety threshold, following the modeling of that distance.

$$d_{p,q}=\sqrt{{\left(x_p-x_q\right)}^2+{\left(y_p-y_q\right)}^2+{\left(z_p-z_q\right)}^2}$$

(14)

$$V_b=\left\{\begin{array}{cc}0& d_{p,q}>2d_2\\ d_2+d_1-d_{p,q}& 2d_1<d_{p,q}<2d_2\\ 1000& d_{p,q}<d_1\end{array}\right.$$

(15)

Here, $d_{p,q}$ represents the distance between the $p$-th and the $q$-th drones. In multi-drone collaborative operation scenarios, as shown in Equations (14) and (15), the path planning algorithm for each drone invokes this model in real time during actual calculations to obtain distance information relative to other drones $d_{p,q}$. If $d_{p,q}$ approaches or falls below the safety threshold, a large penalty value will be assigned in accordance with Equation (15). The model adjusts the drone’s waypoints by combining the exploration and exploitation mechanisms of the Crocodile Ambush Optimization Algorithm (CAOA). In multi-UAV cooperative missions, each aircraft performs individual path optimization. A spatial separation penalty term is embedded in the cost function to restrict inter-UAV distance within a safe range and achieve effective collision avoidance. The strategy maintains flight safety while retaining high computational efficiency, making it well adaptable to real engineering deployment.

3.2.6. Constraints

Constraints can be used to control drone flight within a specified airspace. To set constraints:

$$\left\{\begin{array}{c}0\le x_i\le x_{\max }\\ 0\le y_i\le y_{\max }\\ 0\le z_i\le z_{\max }\end{array}\right.,i=1,2,3,\cdot \cdot \cdot ,n$$

(16)

$x_{\max }$, $y_{\max }$, $z_{\max }$. This value can be adjusted as needed; in the experiments described in this paper, it was set to 1000 throughout.

3.2.7. Objective Function

UAV path planning can be formulated as a constrained single-objective or multi-objective optimization problem. This paper focuses on searching for a unique optimal trajectory, whereas multi-objective optimization is designed to balance two or more conflicting objectives, which does not match the research scope of this work. Accordingly, multiple constraints, including flight range, terrain and obstacle limitations, flight altitude, and pitch angle, are integrated into a unified single-objective cost function via a weighted fusion strategy. Nevertheless, a systematic sensitivity analysis of the weighting parameters is not conducted in this study, which will be investigated comprehensively in our future research. The multi-constrained optimization problem is further transformed into a weighted single-objective optimization model, which is also the mainstream modeling approach in UAV trajectory planning. This method not only satisfies practical physical constraints but also guarantees the computational efficiency of trajectory optimization. The constructed objective function comprises three parts: total flight distance, obstacle avoidance cost, and flight cost within the airspace boundary. The specific mathematical expression is presented as follows:

$$W=\min \left(w_1{V}_c+w_2{V}_s+w_3{V}_h+w_4{V}_f+w_5{V}_b\right)$$

(17)

$$w_1+w_2+w_3+w_4+w_5=1$$

(18)

In the equation: $V_c$ represents the total flight cost of the UAV, with a weight of $w_1$; $V_s$ represents the cost of the UAV flying around obstacles, with a weight of $w_2$; $V_h$ represents the cost of the UAV flying within the specified altitude, with a weight of $w_3$; $V_f$ is the cost of the pitch angle constraint, with a weight of $w_4$; $V_b$ is the cost of the spatial boundary constraint, with a weight of $w_5$.

4. Crocodile Ambush Optimization Algorithm (CAOA)

The Crocodile Ambush Optimization Algorithm (CAOA) is a novel meta-heuristic optimization method inspired by the ambush hunting strategy of crocodiles. It effectively mimics the crocodile’s hunting mechanism through an adaptive decay model of the crocodile’s energy state, a population leader strategy based on random selection, and a threshold-based reinitialization mechanism triggered when the search stalls. The algorithm framework and pseudocode are shown in Figure 3.

Figure 3. Flowchart of the Crocodile Ambush Optimization Algorithm.

The algorithm can be divided into three stages.

4.1. Population Initialization and Global Search

A random initial population is generated within the search space. Each crocodile selects the optimal individual as the population leader based on an evaluation of the current position’s suitability using its fitness value $f\left(x_i\right)$; crocodiles with lower fitness values have a higher probability of becoming leaders, thereby achieving an initial global exploration of the entire solution space:

$$x_{i,j}=l{b}_j+r·\left(u{b}_j-l{b}_j\right)$$

(19)

Let $x_{i,j}$ be the $i$-th crocodile at the position in the $j$-th dimension, where $l{b}_j$ and $u{b}_j$ denote the upper and lower bounds of the search space in the $j$-th dimension, respectively, and $r$ is a random number uniformly distributed on [0, 1]. The probability model for selecting the $i$-th crocodile as the leader is defined as follows:

$$P\left(x_i\right)={\displaystyle \frac{1}{1+f\left(x_i\right)}}$$

(20)

4.2. Energy Dynamics and Ambush Selection

By controlling the search intensity of individual agents through a mechanism in which energy adaptively decays with distance traveled, and by combining this with a fitness-based probabilistic leader selection strategy, detailed local exploration can be performed near high-quality regions while preventing the algorithm from premature convergence.

To simulate this behavior, each crocodile agent is assigned an energy value. Since all crocodiles should start in the same initial state, the expression is given as follows:

$$E_i=E_{init},{\forall }_i\in \left\{1,2,3,\dots ,N\right\}$$

(21)

where $E_{init}$ is a predefined constant, and N is the number of crocodiles in the population. The algorithm’s update rule is as follows:

$$E_i^{t+1}=E_i^t-\gamma ·‖\Delta x_i‖$$

(22)

Here, $E_i^{t+1}$ and $E_i^t$ denote the energy levels of the $i$-th crocodile at the $\left(t+1\right)$-th and $t$-th iterations, respectively. The parameter $\gamma$ is the energy consumption coefficient, and $\left|\Delta x_i\right|$ represents the Euclidean distance traveled by the crocodile in the current iteration, the calculation formula is.

$$‖\Delta x_i‖=\sqrt{{{\displaystyle {\sum }_{j=1}^n\left(x_{i,j}^{t+1}-x_{i,j}^t\right)}}^2}$$

(23)

In the equation, $n$ represents the dimension of the search space, $x_{i,j}^{t+1}$ and $x_{i,j}^t$ denote the positions of the $i$-th crocodile in the $j$-th dimension at the $\left(t+1\right)$-th and $t$-th iterations, respectively.

4.3. Search Iteration and Termination

The formula for updating the position of an individual at the $\left(t+1\right)$-th iteration is as follows:

$$x_i^{t+1}=x_i^t+\alpha ·\left(bes{t}_j^t-x_{i,j}^t\right)+\beta ·\left(1-2r\right)$$

(24)

where $x_i^{t+1}$ represents the updated spatial position of the $i$-th crocodile, $x_i^t$ is its current position, $bes{t}_j^t$ is the component of the leader crocodile in the $t$-th dimension at the $j$-th iteration, $\alpha$ is the step size parameter that controls the extent of the leader’s influence, $\beta$ is the perturbation intensity coefficient, and $r$ is a random number uniformly distributed on the interval [0, 1].

By monitoring fluctuations in the objective function value $f\left(x\right)$ and dynamically adjusting the positions of the individuals based on significant improvements or deteriorations in solution quality, CAOA calculates the change in the objective function value for each crocodile individual between consecutive iterations using Equation (25):

$$\Delta f\left(x_i\right)=f_{current}\left(x_i\right)-f_{previous}\left(x_i\right)$$

(25)

Here, $f_{current}\left(x_i\right)$ represents the objective function value of the $i$-th crocodile in the current iteration, and $f_{previous}\left(x_i\right)$ represents the objective function value of the same crocodile in the previous iteration. $\Delta f\left(x_i\right)$ is used to quantify the magnitude and direction of the change in quality.

The algorithm continuously evaluates and updates itself during the iterative loop, ultimately obtaining the global optimal solution $x_{global}$, which is mathematically defined as:

$$x_{global}=\arg \underset{x_i\in P}{\min } f\left(x_i\right)$$

(26)

where $P=\left\{x_1,x_2,x_3,\dots ,x_N\right\}$ denotes the set of positions of all crocodiles in the population, $f\left(x_i\right)$ is the objective function value corresponding to the position of the $i$-th crocodile, and $x_{global}$ represents the optimal position in the population with the minimum objective function value.

4.4. CAOA Pseudo-Code

In summary, the pseudo-code of the CAOA part of the algorithm flow is shown in Algorithm 1.

Algorithm 1 CAOA pseudo-code

Input: N,iter, lb,ub,dim,fobj Output: gBestScore, gBest,cg_curve

01: Initialize population positions; 02: Set initial energies Ei = E0, compute initial fitness f(xi); 03: Set global best gBest and gBestScore; 04: for t = 1 to T do 05:      Store previous positions and fitness values; 06:      Select leader based on inverse fitness probability; 07:      for i = 1 to N do 08:            if i is leader then; 09:                  Continue; 10:            end if 11:            Generate random vector r; 12:            Update position: xi ← xi + α(x_leader − xi) + β(1 − 2r); 13:            Apply boundary control to xi; 14:            Compute new fitness f_new = f(xi); 15:            if |f_new − f_old| > δ and f_new > f_old then 16:                   Reinitialize xi randomly within bounds; 17:                   Recompute f_new; 18:            end if 19:            Update xi and f(xi); 20:            Compute energy decay: Ei ← Ei − γ · distance(xi, xi^old); 21:            If Ei ≤ 0 then 22:                   Reinitialize xi and Ei; Recompute fitness; 23:            end if 24:            Update global best if better solution found; 25:      end for 26:      Record gBestScore in convergence curve; 27: end for 28: return gBest, gBestScore, convergence history;

5. HPSOCAOA Optimization Algorithm

First, a collaborative search framework combining the Particle Swarm Optimization (PSO) algorithm and the Crocodile Ambush Algorithm (CAOA) is established; Second, an adaptive energy consumption coefficient is designed to address the problems of premature individual elimination in the early stages of the algorithm and insufficient convergence momentum in the later stages, thereby further balancing global exploration and local exploitation. Finally, cross-learning is introduced through a mechanism for replacing individuals across groups; high-quality individuals identified by PSO’s fine-tuning are selectively substituted for poorer-performing individuals in CAOA. This resolves the problems of redundant low-quality individuals within the population and low search efficiency, thereby enhancing the overall optimization performance. An effective balance between global exploration and local exploitation is achieved, with excellent convergence speed and high solution accuracy integrated. The algorithm framework and pseudocode are shown in Figure 4.

Figure 4. Flowchart of the Hybrid PSOCAOA Optimization Algorithm.

5.1. PSO-CAOA Collaborative Search Mechanism

In the Crocodile Ambush Optimization Algorithm, the initial population consists of uniformly distributed random numbers. To address the imbalance between global search and local exploration capabilities in this algorithm, we combine the powerful local exploration of the Particle Swarm Optimization (PSO) algorithm with the efficient global exploration of the Crocodile Ambush Optimization Algorithm to construct a collaborative search mechanism between the two. The entire population is divided into a CAOA global exploration subpopulation and a PSO local development subpopulation. The two subpopulations evolve independently, complementing each other’s strengths to jointly complete the optimization search.

(1)

Global Exploration by the CAOA Subpopulation

The CAOA subpopulation comprises 70% of the initial population, and Equation (24) serves as its position update formula. This configuration ensures extensive spatial coverage and maintains population diversity.

(2)

Local Exploration by the PSO Subpopulation

The PSO subpopulation comprises 30% of the initial population. PSO individuals perform fine-grained searches around their own historical optimal solutions and the global optimal solution. The velocity and position update formulas are

$$V_{i,d}(t+1)=w{V}_{i,d}(t)+c_1{r}_1(p_{i,d}-x_{i,d}(t))+c_2{r}_2(g_d-x_{i,d}(t))$$

(27)

$$x_{i,d}(t+1)=x_{i,d}(t)+V_{i,d}(t+1)$$

(28)

Here, $V_{i,d}$ represents the velocity of the $i$-th particle in the $d$-th dimension, $x_{i,d}(t)$ represents the position of the $i$-th particle in the $d$-th dimension, $w$ is the inertial weight, $c_1,c_2$ is the learning factor, $r_1,r_2$ is a random number uniformly distributed in the interval [0, 1], $p_{i,d}$ is the individual historical optimal position of the $i$-th particle, and $g_d$ is the global optimal position of the entire population. PSO offers the advantages of a simple structure, fast convergence, and strong local optimization capabilities. Combining the two approaches allows them to complement each other’s strengths.

5.2. Adaptive Energy Consumption Mechanism

Considering that the original CAOA adopts a fixed constant energy coefficient, it lacks dynamic adjustment ability during the whole iteration and exhibits poor adaptability to global exploration in the early stage and to local exploitation in the later stage. The fixed energy coefficient cannot dynamically match the search state of the population in the iterative path planning process, which usually leads to overly fast energy consumption at the beginning of the iteration and excessively slow consumption in the later period, making it difficult to balance global exploration and local exploitation. The linear strategy is a widely used improvement idea in evolutionary algorithms; for example, the improved schemes in References [8,18] both adopt the linear adjustment method and achieve satisfactory performance. Meanwhile, compared with complex nonlinear energy decay strategies, the linear adaptive form features a simple structure, strong interpretability, and low computational cost, which is more suitable for the real-time requirements of UAV path planning in this study.

To address these issues, an energy consumption coefficient that linearly adapts with the number of iterations has been designed. This allows for slower energy decay during the early iterations, reducing the population’s reset rate, preserving greater population diversity, and enhancing global exploration capabilities. In later iterations, energy decay is accelerated to speed up the elimination of low-quality individuals, thereby strengthening exploitation and convergence performance. The expression is as follows:

$$\gamma \left(t\right)={\gamma }_{\min }+\left({\gamma }_{\max }-{\gamma }_{\min }\right)·{\displaystyle \frac{t}{T_{\max }}}$$

(29)

where $\gamma \left(t\right)$ is the energy consumption coefficient for the $t$-th iteration, ${\gamma }_{\min }$ and ${\gamma }_{\max }$ are the lower and upper bounds of the energy consumption coefficient, respectively; $T_{\max }$ is the maximum number of iterations. After an individual’s position is updated, its energy decays according to Equation (22); when an individual’s energy falls below the threshold, its position is reset.

5.3. Cross-Disciplinary Learning

To address the issues of poor information exchange between the PSO subpopulation and the CAOA subpopulation, low search efficiency caused by redundant, low-quality individuals within the populations, and the tendency to get stuck in local optima, we introduce cross-learning based on the collaborative search architecture. This approach directs high-quality development outcomes from the PSO subpopulation back to the CAOA subpopulation while simultaneously eliminating low-quality individuals from the CAOA subpopulation. This makes the algorithm’s global exploration more targeted. This enhances search efficiency, facilitates information exchange between the two groups, improves population quality, and further strengthens the algorithm’s overall optimization performance. The specific process is as follows:

(1)

After each iteration, calculate the fitness values of all individuals in both the PSO subpopulation and the CAOA subpopulation.

(2)

Select the top individuals with the highest fitness from the PSO subpopulation and replace them one-to-one with the lowest-quality individuals identified in the CAOA subpopulation. This completes the population update, and the process proceeds to the next iteration.

The number of replacements $k$ is calculated using the following formula.

$$k=round\left(N_{CAOA}\times \eta \right)$$

(30)

In the equation, $N_{CAOA}$ represents the size of the CAOA subpopulation, and $\eta$ represents the replacement ratio coefficient; in this paper, $\eta =0.15$. The core advantage of this mechanism is that it achieves a precise integration of local exploitation of high-quality information with global exploration. It not only improves the overall quality of the CAOA subpopulation but also guides CAOA to focus its search on high-quality regions in subsequent iterations, thereby enhancing the synergy between PSO and CAOA. This improves the optimization accuracy and convergence stability.

5.4. HPSOCAOA Pseudo-Code

In summary, the pseudo-code of the HPSOCAOA part of the algorithm flow is shown in Algorithm 2.

Algorithm 2 HPSOCAOA pseudo-code

Input: N,iter, lb,ub,dim,fobj Output: gBestScore, gBest,cg_curve

01: Initialize algorithm parameters; Initialize population positions; 02: Split population into CAOA and PSO sub-populations; 03: Set initial CAOA energies Ei = E0, initialize PSO velocities vi and personal bests pBesti; Compute initial fitness f(xi); 04: Set global best gBest and gBestScore; Set cross-group replacement ratio; 05: for t = 1 to T do 06:      Store previous positions and fitness values; 07:      Calculate adaptive energy consumption coefficient γ(t); 08:      Select CAOA leader based on inverse fitness probability; 09:      for i = 1 to N_CAOA do 10:            if i is leader then; 11:                  Continue; 12:            end if 13:            Generate random vector r; 14:            Update position: xi ← xi + α(x_leader − xi) + β(1 − 2r); 15:            Apply boundary control to xi; 16:            Compute new fitness f_new = f(xi); 17:            if |f_new − f_old| > δ and f_new > f_old then 18:                   Reinitialize xi randomly within bounds; 19:                   Recompute f_new; 20:            end if 21:            Update xi and f(xi); 22:            Compute energy decay: Ei ← Ei − γ(t) · distance(xi, xiᵒˡᵈ); 23:            If Ei ≤ 0 then 24:                   Reinitialize xi and Ei; Recompute fitness; 25:            end if 26:            Update global best if better solution found; 27:      end for 28:      Calculate adaptive ω(t), c1(t), c2(t) for PSO; 29:      Update PSO sub-population velocities and positions; Update pBesti; 30:      Execute cross-group elite replacement: PSO best → CAOA worst; 31:      Update global best gBest and gBestScore; 32:      Record gBestScore in convergence curve; 33: end for 34: return gBest, gBestScore, convergence history;

As shown in line 02 of the pseudocode table, the two-population partitioning strategy is introduced during the population initialization phase. The total population is divided into a CAOA subpopulation responsible for global exploration and a PSO subpopulation responsible for local exploitation in a certain ratio; this strategy effectively balances the algorithm’s exploration and exploitation capabilities, thereby improving the overall optimization efficiency and convergence accuracy. In lines 07 and 22 of the code table, the energy decay mechanism is dynamically adjusted by introducing an adaptive energy consumption coefficient, allowing individual energy consumption to be adaptively changed with the iteration process. This enhances the algorithm’s stability across different search phases and prevents the premature decline of population vitality. Lines 28–29 of the code table demonstrate the PSO adaptive parameter tuning mechanism. By dynamically calculating the adaptive inertia weight and the adaptive learning factors, particles are enabled to autonomously balance global and local search capabilities during the iteration process, which significantly improves the optimization accuracy and reduces the risk of being trapped in local optima. Finally, as shown in line 30 of the code table, a cross-population elite replacement strategy is introduced. The worst individual in the CAOA subpopulation is replaced with the best individual from the PSO subpopulation; through the sharing of high-quality solution information and the promotion of cooperative optimization among populations, global convergence is effectively accelerated, and the quality of optimal solutions is improved.

6. Algorithm Testing

6.1. Parameter Sensitivity Analysis

To investigate the impact of key parameters on the performance of the proposed HPSOCAOA algorithm, a sensitivity analysis was conducted. Two core parameters were selected: the CAOA population allocation ratio (CAOA ratio) and the replacement rate of inferior CAOA individuals (replacement rate). Each parameter is varied within a reasonable interval, while the remaining parameters are set according to the original reference. The standard 30-dimensional Sphere function was used as the benchmark, and the algorithm was run independently for each parameter setting. The results show that the overall performance of the algorithm is optimal when the CAOA ratio = 0.7 and the replacement rate = 0.15. Specifically, a larger CAOA population allocation ratio is not conducive to enhancing exploration ability, and improving population quality is more effective than simply increasing population size in promoting the generation of high-quality leaders in the crocodile population, as shown in Figure 5.

Figure 5. Parameter sensitivity analysis for the CAOA ratio and replacement rate. Each curve shows the mean ± standard deviation over 10 runs.

6.2. Convergence Analysis

To verify the effectiveness of the proposed Hybrid Particle Swarm and Crocodile Ambush Optimization Algorithm (HPSOCAOA), the Particle Swarm Optimization (PSO) algorithm, LSHADE, LSHADEEpSin, LSHADESPACMA, and the Crocodile Ambush Optimization Algorithm (CAOA) are selected as comparative algorithms. Comparative experiments are conducted on 29 standard test functions, where the adopted test functions are the CEC2017 benchmark suite from Reference [24], as listed in Table 2. Function F2 is excluded due to convergence issues. Among the remaining functions, F1 and F3 are unimodal functions, which are mainly adopted to evaluate the convergence speed and optimization accuracy of the algorithms. Functions F4 to F10 are complex multimodal functions, which are primarily utilized to test the global exploration capability of the algorithms. Functions F11 to F30 are constructed by combining unimodal and multimodal components, which can comprehensively evaluate the solution accuracy and stability of each algorithm for different optimization problems.

Table 2. CEC 2017 Test Set.

Letter: Number	Function Name	Optimal Value
F1	Shifted and Rotated Bent Cigar Function	100
F3	Shifted and Rotated Zakharov Function	300
F4	Shifted and Rotated Rosenbrock’s Function	400
F5	Shifted and Rotated Rastrigin’s Function	500
F6	Shifted and Rotated Expanded Scaffer’s F6 Function	600
F7	Shifted and Rotated Lunacek Bi_Rastrigin Function	700
F8	Shifted and Rotated Non-Continuous Rastrigin’s Function	800
F9	Shifted and Rotated Levy Function	900
F10	Shifted and Rotated Schwefel’s Function	1000
F11	Hybrid Function 1 (N = 3)	1100
F12	Hybrid Function 2 (N = 3)	1200
F13	Hybrid Function 3 (N = 4)	1300
F14	Hybrid Function 4 (N = 4)	1400
F15	Hybrid Function 5 (N = 4)	1500
F16	Hybrid Function 6 (N = 5)	1600
F17	Hybrid Function 7 (N = 5)	1700
F18	Hybrid Function 8 (N = 5)	1800
F19	Hybrid Function 9 (N = 5)	1900
F20	Hybrid Function 10 (N = 6)	2000
F21	Composition Function 1 (N = 3)	2300
F22	Composition Function 2 (N = 3)	2200
F23	Composition Function 3 (N = 4)	2300
F24	Composition Function 4 (N = 4)	2400
F25	Composition Function 5 (N = 5)	2500
F26	Composition Function 6 (N = 5)	2600
F27	Composition Function 7 (N = 6)	2700
F28	Composition Function 8 (N = 6)	2800
F29	Composition Function 9 (N = 3)	2900
F30	Composition Function 10 (N = 3)	3000

To ensure experimental fairness, all experiments are performed on a computer equipped with a 64-bit Windows 10 operating system and 16 GB of RAM, with MATLAB R2024a used for simulation. For all comparative algorithms, the population size is set to 30, the maximum number of iterations is set to 500, and each test function is run independently for 30 times. The mean (Mean) and standard deviation (Std) of the optimal solutions obtained by each algorithm are calculated. The parameter settings of the six algorithms are shown in Table 3, and the experimental results are presented in Table 4, Table 5 and Table 6.

Table 3. Compare the relevant parameters of algorithms.

Algorithms	Parameters
HPSOCAOA	$\alpha =0.3$, $\beta =0.1$, $\delta ={10}^{-4}$, $w=0.7, c_1=2.05, c_2=2.05$
CAOA	$\alpha =0.3$, $\beta =0.1$, $\gamma =0.1$, $\delta ={10}^{-4}$
PSO	$w=0.7, c_1=2.05, c_2=2.05$
LSHADE	$p=0.11, H=5, F=0.5, CR=0.5$
LSHADEEpSin	$p=0.11, H=5, F=0.5, CR=0.5, pb=0.4, ps=0.5, f_{init}=0.5$
LSHADESPACMA	$c=0.8, Fcp=0.5, arc\_rate=1.4, p=0.11, H=5, F=0.5, CR=0.5, \min popsize=4$

Table 4. Test Results.

Fun	Index	HPSOCAOA	CAOA	PSO	LSHADE	LSHADEEpSin	LSHADESPACMA
F1	Mean	1.506 × 104	1.297 × 1010	1.586 × 1010	7.199 × 107	2.194 × 105	8.533 × 107
	Std	1.164 × 104	1.547 × 109	5.728 × 109	2.057 × 108	3.549 × 105	2.117 × 108
	Min	1.283 × 103	1.030 × 1010	7.061 × 109	2.017 × 105	5.805 × 103	2.902 × 105
F3	Mean	3.775 × 104	5.212 × 104	1.351 × 105	2.028 × 105	2.133 × 104	4.579 × 104
	Std	1.027 × 104	8.128 × 103	2.486 × 104	8.947 × 104	9.650 × 103	2.453 × 104
	Min	1.406 × 104	3.529 × 104	8.022 × 104	7.213 × 104	7.318 × 103	1.634 × 104
F4	Mean	5.202 × 102	2.982 × 103	2.183 × 103	4.599 × 102	4.998 × 102	5.203 × 102
	Std	4.738 × 101	4.909 × 102	5.580 × 102	3.537 × 101	2.095 × 101	2.055 × 101
	Min	4.096 × 102	1.748 × 103	1.412 × 103	4.281 × 102	4.650 × 102	4.727 × 102
F5	Mean	6.629 × 102	7.206 × 102	8.084 × 102	6.651 × 102	5.921 × 102	6.423 × 102
	Std	3.788 × 101	2.972 × 101	3.452 × 101	6.031 × 101	2.946 × 101	2.903 × 101
	Min	5.786 × 102	6.656 × 102	7.381 × 102	5.604 × 102	5.429 × 102	5.956 × 102
F6	Mean	6.345 × 102	6.570 × 102	6.520 × 102	6.019 × 102	6.024 × 102	6.087 × 102
	Std	7.762 × 100	6.525 × 100	8.579 × 100	3.646 × 100	1.372 × 100	4.309 × 100
	Min	6.213 × 102	6.470 × 102	6.351 × 102	6.000 × 102	6.007 × 102	6.014 × 102
F7	Mean	9.486 × 102	1.079 × 103	1.233 × 103	9.128 × 102	8.528 × 102	9.662 × 102
	Std	6.676 × 101	7.662 × 101	8.440 × 101	6.032 × 101	3.369 × 101	6.335 × 101
	Min	8.378 × 102	9.263 × 102	1.070 × 103	7.851 × 102	7.857 × 102	8.689 × 102
F8	Mean	9.320 × 102	9.620 × 102	1.087 × 103	9.719 × 102	8.994 × 102	9.336 × 102
	Std	3.595 × 101	3.576 × 101	2.676 × 101	4.851 × 101	2.499 × 101	2.172 × 101
	Min	8.696 × 102	8.926 × 102	1.035 × 103	8.678 × 102	8.569 × 102	9.033 × 102
F9	Mean	2.467 × 103	4.936 × 103	9.457 × 103	9.821 × 102	1.520 × 103	3.027 × 103
	Std	4.938 × 102	1.005 × 103	2.284 × 103	7.531 × 101	4.857 × 102	1.389 × 103
	Min	1.621 × 103	3.312 × 103	4.661 × 103	9.025 × 102	9.704 × 102	1.238 × 103
F10	Mean	4.891 × 103	6.245 × 103	8.501 × 103	8.273 × 103	5.977 × 103	5.404 × 103
	Std	5.756 × 102	9.245 × 102	5.688 × 102	6.104 × 102	8.895 × 102	4.701 × 102
	Min	3.875 × 103	4.418 × 103	7.319 × 103	6.758 × 103	4.190 × 103	4.060 × 103
F11	Mean	1.278 × 103	2.072 × 103	2.797 × 103	1.506 × 103	1.289 × 103	1.292 × 103
	Std	7.225 × 101	1.811 × 102	1.046 × 103	6.649 × 102	5.424 × 101	7.882 × 101
	Min	1.154 × 103	1.756 × 103	1.646 × 103	1.188 × 103	1.191 × 103	1.188 × 103
F12	Mean	2.999 × 106	2.085 × 109	8.359 × 108	3.866 × 107	2.877 × 105	1.064 × 106
	Std	3.740 × 106	4.150 × 108	3.842 × 108	1.205 × 108	3.914 × 105	8.513 × 105
	Min	2.388 × 105	1.553 × 109	1.572 × 108	8.489 × 105	2.380 × 104	8.384 × 104
F13	Mean	2.959 × 104	2.929 × 108	2.116 × 107	2.960 × 105	1.796 × 104	3.503 × 104
	Std	2.702 × 104	2.274 × 108	2.955 × 107	5.912 × 105	1.696 × 104	2.295 × 104
	Min	6.452 × 103	1.028 × 107	3.784 × 105	1.569 × 104	2.365 × 103	9.517 × 103
F14	Mean	5.433 × 104	4.097 × 105	8.127 × 105	6.934 × 104	1.714 × 103	7.255 × 104
	Std	3.972 × 104	2.298 × 105	1.910 × 106	1.254 × 105	4.984 × 102	1.700 × 105
	Min	5.628 × 103	4.497 × 104	4.156 × 104	2.140 × 103	1.485 × 103	1.494 × 103
F15	Mean	7.966 × 103	3.171 × 104	8.580 × 104	1.153 × 105	3.069 × 103	4.074 × 103
	Std	7.876 × 103	1.553 × 104	3.211 × 105	1.441 × 105	1.913 × 103	2.043 × 103
	Min	1.928 × 103	1.139 × 104	4.489 × 103	3.113 × 103	1.813 × 103	1.824 × 103
F16	Mean	2.836 × 103	3.879 × 103	3.792 × 103	3.403 × 103	2.385 × 103	2.911 × 103
	Std	4.139 × 102	4.726 × 102	3.350 × 102	4.764 × 102	2.994 × 102	1.827 × 102
	Min	1.905 × 103	3.192 × 103	2.907 × 103	2.135 × 103	1.804 × 103	2.412 × 103
F17	Mean	2.457 × 103	2.616 × 103	2.582 × 103	2.475 × 103	1.940 × 103	2.095 × 103
	Std	2.619 × 102	2.594 × 102	2.773 × 102	2.784 × 102	1.011 × 102	1.292 × 102
	Min	1.808 × 103	1.937 × 103	1.983 × 103	1.960 × 103	1.796 × 103	1.824 × 103
F18	Mean	7.766 × 105	1.377 × 106	7.564 × 106	2.798 × 106	3.958 × 104	1.954 × 105
	Std	7.681 × 105	1.050 × 106	7.622 × 106	3.333 × 106	2.186 × 104	3.339 × 105
	Min	3.351 × 104	9.800 × 104	1.248 × 106	1.048 × 105	4.780 × 103	1.250 × 104
F19	Mean	9.225 × 103	1.290 × 106	1.227 × 107	9.647 × 104	2.173 × 103	3.317 × 103
	Std	8.715 × 103	9.281 × 105	2.769 × 107	3.232 × 105	1.803 × 102	1.545 × 103
	Min	2.071 × 103	2.232 × 104	1.520 × 104	2.301 × 103	1.984 × 103	2.039 × 103
F20	Mean	2.553 × 103	2.619 × 103	2.672 × 103	2.650 × 103	2.309 × 103	2.565 × 103
	Std	2.278 × 102	1.876 × 102	2.348 × 102	3.016 × 102	1.334 × 102	1.272 × 102
	Min	2.110 × 103	2.271 × 103	2.286 × 103	2.099 × 103	2.099 × 103	2.292 × 103
F21	Mean	2.462 × 103	2.554 × 103	2.589 × 103	2.481 × 103	2.405 × 103	2.440 × 103
	Std	4.420 × 101	4.941 × 101	3.162 × 101	4.814 × 101	2.496 × 101	3.329 × 101
	Min	2.383 × 103	2.489 × 103	2.521 × 103	2.355 × 103	2.354 × 103	2.388 × 103
F22	Mean	2.956 × 103	6.108 × 103	8.219 × 103	8.474 × 103	3.260 × 103	2.342 × 103
	Std	1.499 × 103	1.545 × 103	2.529 × 103	1.896 × 103	1.930 × 103	3.803 × 101
	Min	2.300 × 103	3.847 × 103	3.322 × 103	2.960 × 103	2.301 × 103	2.315 × 103
F23	Mean	2.986 × 103	3.302 × 103	3.001 × 103	2.831 × 103	2.755 × 103	2.790 × 103
	Std	1.221 × 102	1.113 × 102	2.689 × 101	5.600 × 101	2.949 × 101	2.454 × 101
	Min	2.799 × 103	3.080 × 103	2.937 × 103	2.728 × 103	2.712 × 103	2.748 × 103
F24	Mean	3.198 × 103	3.544 × 103	3.152 × 103	3.012 × 103	2.930 × 103	2.985 × 103
	Std	1.440 × 102	1.236 × 102	1.919 × 101	5.609 × 101	3.780 × 101	3.450 × 101
	Min	2.905 × 103	3.339 × 103	3.096 × 103	2.923 × 103	2.866 × 103	2.924 × 103
F25	Mean	2.912 × 103	3.196 × 103	3.675 × 103	2.897 × 103	2.898 × 103	2.923 × 103
	Std	2.488 × 101	3.587 × 101	3.035 × 102	1.404 × 101	1.468 × 101	2.050 × 101
	Min	2.884 × 103	3.137 × 103	3.232 × 103	2.879 × 103	2.884 × 103	2.889 × 103
F26	Mean	5.641 × 103	8.313 × 103	7.011 × 103	5.189 × 103	4.698 × 103	4.396 × 103
	Std	1.788 × 103	5.164 × 102	6.383 × 102	5.629 × 102	3.695 × 102	9.706 × 102
	Min	2.801 × 103	7.313 × 103	5.228 × 103	3.998 × 103	3.984 × 103	2.901 × 103
F27	Mean	3.334 × 103	4.008 × 103	3.350 × 103	3.200 × 103	3.235 × 103	3.239 × 103
	Std	8.014 × 101	1.995 × 102	4.554 × 101	1.657 × 104	1.918 × 101	1.923 × 101
	Min	3.233 × 103	3.581 × 103	3.269 × 103	3.200 × 103	3.201 × 103	3.215 × 103
F28	Mean	3.276 × 103	4.198 × 103	4.220 × 103	3.299 × 103	3.260 × 103	3.300 × 103
	Std	4.988 × 101	1.119 × 102	3.485 × 102	2.487 × 100	4.173 × 101	4.516 × 101
	Min	3.209 × 103	3.961 × 103	3.714 × 103	3.289 × 103	3.213 × 103	3.239 × 103
F29	Mean	4.196 × 103	5.176 × 103	4.775 × 103	4.187 × 103	3.729 × 103	3.797 × 103
	Std	2.632 × 102	3.691 × 102	2.520 × 102	4.390 × 102	1.728 × 102	1.551 × 102
	Min	3.770 × 103	4.528 × 103	4.319 × 103	3.402 × 103	3.471 × 103	3.585 × 103
F30	Mean	4.251 × 104	1.477 × 107	5.987 × 106	4.551 × 104	2.468 × 104	5.623 × 104
	Std	3.727 × 104	7.154 × 106	7.528 × 106	1.041 × 105	4.154 × 104	3.554 × 104
	Min	9.773 × 103	1.381 × 106	5.693 × 105	4.340 × 103	7.513 × 103	1.689 × 104
Time(s)	Mean	16.12	14.49	11.94	26.76	13.52	13.26

Table 5. Wilcoxon Rank-Sum Test.

Fun	CAOA	PSO	LSHADE	LSHADE EpSin	LSHADE SPACMA
F1	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	4.686 × 10−8	3.020 × 10−11
F3	1.108 × 10−6	3.020 × 10−11	3.020 × 10−11	1.873 × 10−7	8.650 × 10−1
F4	3.020 × 10−11	3.020 × 10−11	2.879 × 10−6	3.778 × 10−2	6.843 × 10−1
F5	3.805 × 10−7	3.020 × 10−11	8.073 × 10−1	1.202 × 10−8	2.236 × 10−2
F6	4.504 × 10−11	5.967 × 10−9	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11
F7	9.833 × 10−8	3.690 × 10−11	9.926 × 10−2	3.646 × 10−8	2.905 × 10−1
F8	2.052 × 10−3	3.020 × 10−11	8.564 × 10−4	2.839 × 10−4	9.234 × 10−1
F9	3.690 × 10−11	3.020 × 10−11	3.020 × 10−11	3.965 × 10−8	4.290 × 10−1
F10	6.526 × 10−7	3.020 × 10−11	3.020 × 10−11	6.736 × 10−6	4.459 × 10−4
F11	3.020 × 10−11	3.020 × 10−11	3.403 × 10−1	3.555 × 10−1	4.464 × 10−1
F12	3.020 × 10−11	3.020 × 10−11	7.199 × 10−5	6.528 × 10−8	4.841 × 10−2
F13	3.020 × 10−11	3.020 × 10−11	7.043 × 10−7	6.377 × 10−3	1.809 × 10−1
F14	2.227 × 10−9	1.698 × 10−8	2.921 × 10−2	3.020 × 10−11	9.514 × 10−6
F15	6.518 × 10−9	4.801 × 10−7	2.015 × 10−8	3.831 × 10−5	9.049 × 10−2
F16	3.820 × 10−10	5.573 × 10−10	1.019 × 10−5	2.773 × 10−5	7.283 × 10−1
F17	3.265 × 10−2	1.494 × 10−1	8.187 × 10−1	5.573 × 10−10	1.067 × 10−7
F18	5.828 × 10−3	5.573 × 10−10	1.695 × 10−2	1.777 × 10−10	2.491 × 10−6
F19	4.077 × 10−11	1.464 × 10−10	3.032 × 10−2	2.783 × 10−7	7.617 × 10−3
F20	2.973 × 10−1	4.515 × 10−2	3.953 × 10−1	1.430 × 10−5	5.793 × 10−1
F21	1.429 × 10−8	6.066 × 10−11	9.926 × 10−2	8.197 × 10−7	4.841 × 10−2
F22	1.596 × 10−7	1.411 × 10−9	1.329 × 10−10	2.133 × 10−5	9.514 × 10−6
F23	5.072 × 10−10	1.858 × 10−1	2.572 × 10−7	4.504 × 10−11	1.329 × 10−10
F24	2.371 × 10−10	1.580 × 10−1	8.197 × 10−7	1.070 × 10−9	1.067 × 10−7
F25	3.020 × 10−11	3.020 × 10−11	2.813 × 10−2	1.055 × 10−1	5.369 × 10−2
F26	8.101 × 10−10	3.770 × 10−4	1.564 × 10−2	2.624 × 10−3	2.891 × 10−3
F27	3.020 × 10−11	6.353 × 10−2	3.020 × 10−11	5.462 × 10−9	1.558 × 10−8
F28	3.020 × 10−11	3.020 × 10−11	1.729 × 10−6	1.055 × 10−1	6.377 × 10−3
F29	7.389 × 10−11	3.825 × 10−9	7.618 × 10−1	5.462 × 10−9	3.646 × 10−8
F30	3.020 × 10−11	3.020 × 10−11	1.518 × 10−3	3.157 × 10−5	4.841 × 10−2

Table 6. Friedman Statistical Results.

Fun	HPSOCAOA	CAOA	PSO	LSHADE	LSHADE EpSin	LSHADE SPACMA
Mean Rank	2.70	5.03	5.37	3.50	1.53	2.87

To intuitively present the test results of the algorithm, four standard benchmark functions were selected from the four categories of benchmark functions for visualization analysis, as shown in Figure 6. The results highlight the advantages of HPSOCAOA in terms of robustness and convergence accuracy. As shown in Table 4, HPSOCAOA exhibits strong search capabilities, with an average F1 score of 1.506 × 10⁴ and a standard deviation of 1.164 × 10⁴, indicating relatively stable performance. In contrast, CAOA and PSO performed poorly on this function, with mean values of 1.297 × 10¹⁰ and 1.586 × 10¹⁰, respectively. LSHADE, LSHADEEpSin, and LSHADESPACMA achieve relatively better results, with average values of 7.199 × 10⁷, 2.194 × 10⁵, and 8.533 × 10⁷. HPSOCAOA also outperformed the comparison algorithms in F4 and F10, particularly in the F10 function, where its mean of 4.891 × 10³ was significantly lower than the results of other algorithms. This indicates that HPSOCAOA possesses a distinct advantage in local search capabilities for this class of functions.

Figure 6. Convergence Curves Comparison.

In the hybrid function test (F11–F20), the HPSOCAOA algorithm maintains stable and competitive performance. Taking F11 as an example, HPSOCAOA obtains a mean value of 1.278 × 10³, a standard deviation of 7.225 × 10¹, and an optimal value of 1.154 × 10³, which is significantly better than PSO and CAOA. It also achieves a slight advantage over LSHADE, LSHADEEpSin, and LSHADESPACMA. For the remaining composite functions, their results are highly comparable with the other three state-of-the-art algorithms and rank second on several occasions. This indicates that HPSOCAOA performs well in balancing global exploration and local exploitation, and possesses excellent global search capability and strong robustness when addressing complex optimization problems.

In composite functions, HPSOCAOA also possesses certain competitive advantages. Taking F25 as an instance, the mean value of HPSOCAOA is 2.912 × 10³, which is much lower than 3.196 × 10³ of CAOA and 3.675 × 10³ of PSO. Meanwhile, it obtains the smallest standard deviation, reflecting its accurate global search ability in complex solution spaces. In addition, HPSOCAOA presents superior robustness in terms of standard deviation. For the functions where it does not rank first, its performance is very close to the optimal algorithm. This demonstrates that the hybridization mechanism of HPSOCAOA can effectively balance global exploration and local exploitation during the search process. In the entire test set, HPSOCAOA comprehensively outperforms the original CAOA and PSO, which verifies that the adaptive energy consumption coefficient strategy can steadily maintain the quality of the population.

To further evaluate the performance differences between HPSOCAOA and other comparison algorithms, the Wilcoxon rank-sum test is conducted at the significance level of p < 0.05, with HPSOCAOA taken as the reference algorithm. The statistical results are listed in Table 5. It can be observed that there exists a significant difference between HPSOCAOA and the original CAOA on the benchmark set. The results further verify that the proposed improvement strategy has remarkable superiority in enhancing the overall algorithm performance.

Since relying merely on the experimental results in Table 4 and Table 5 is one-sided, the Friedman statistical test is further conducted in this work based on the optimal value, mean value, and standard deviation of each algorithm, and the test results are presented in Table 6. As a classic champion algorithm for the CEC2017 benchmark set, LSHADE-EpSin is endowed with a sine parameter adaptation mechanism and a sophisticated global–local search balance strategy. Excellent adaptability to complex problems and optimization stability are exhibited by this algorithm, which achieves the optimal average ranking and ranks first among all comparative algorithms. Its benchmark superiority in handling high-dimensional, multimodal, and non-convex optimization scenarios is fully highlighted. The average ranking of the proposed HPSOCAOA algorithm is 2.7, ranking second. The overall performance of HPSOCAOA is superior to that of CAOA, WOA, PSO, LSHADE, and LSHADE-SPACMA, and is only inferior to LSHADE-EpSin. Meanwhile, it is indicated that room for performance improvement still exists for the proposed algorithm. In subsequent research, the parameter adaptive regulation idea and dynamic search strategy of LSHADE-EpSin will be referenced. The population update mechanism, parameter adaptive strategy, and local search operator of HPSOCAOA will be further optimized. The performance gap with state-of-the-art competitive algorithms will be narrowed, and the generalization ability and optimization accuracy of the algorithm in high-dimensional constrained optimization and practical engineering applications will be further enhanced.

Further statistical analysis is performed on the runtime of each algorithm. Each benchmark function is run 30 times, and the average runtime of each algorithm is calculated. The results are listed in Table 4. The average runtime of the HPSOCAOA algorithm is 14.49 s, ranking fourth among the six compared algorithms. Its computational time is slightly higher than that of PSO, LSHADE-EpSin, and LSHADE-SPACMA. The proposed algorithm integrates multiple improved modules, including population division, collaborative interaction, and adaptive energy strategy. Compared with conventional single-algorithm approaches, it introduces additional population information interaction and multi-mechanism iterative procedures, which inevitably lead to a slight increase in computational consumption. Nevertheless, the coupling effect of multiple strategies effectively compensates for the search deficiencies of basic algorithms, achieving substantial improvements in solution accuracy and convergence stability. Considering that the proposed algorithm adopts a multi-strategy hybrid evolutionary mechanism, it achieves higher optimization accuracy and convergence performance at the cost of a slight increase in computational complexity. The time overhead is within a reasonable and acceptable range, demonstrating favorable engineering practicability.

In conclusion, all the comparative tests sufficiently validate the prominent performance advantages of the proposed HPSOCAOA algorithm.

7. Simulation Verification and Result Analysis

7.1. Simulation Verification Experiment

To further validate the effectiveness and superiority of the Hybrid Particle Swarm-Crocodile Optimization Algorithm (HPSOCAOA) in practical applications, this paper applies it to the problem of 3D flight-path planning for unmanned aerial vehicles (UAVs). Since there are no published references on the application of CAOA or HPSOCAOA to UAV flight-path planning, we first verify the feasibility of HPSOCAOA in 3D flight-path planning. Four experiments were conducted in a 1000 × 1000 × 1000 environment, with a population size of 100, and a maximum number of iterations of 800. Set the safe distance for drones $d_1=1$, $d_2=3d_1=3$ and the weight settings are mainly determined with reference to the principles reported in Reference [21], and then fine-tuned based on actual engineering experience. Specifically, we appropriately increase the weights of the obstacle-avoidance cost, the pitch angle constraint cost, and the spatial boundary separation cost. This adjustment ensures that the UAV prioritizes safe flight under complex environmental constraints before completing its mission, which is consistent with real-world flight requirements. A unified initialization rule is adopted in all experiments. The number of path waypoints is fixed at 10, and the same random seed is set to control experimental randomness. All comparative algorithms are tested under identical initial conditions, eliminating the interference of random factors on experimental results.

$${\omega }_1=0.10, {\omega }_2=0.35, {\omega }_3=0.15, {\omega }_4=0.25, {\omega }_5=0.15, h_{max}=500, h_{min}=100$$

In a 1000 × 1000 × 1000 three-dimensional space, the start and end points of the UAV are fixed to plan the flight trajectory between them. Three UAVs are planned in a sequential manner. The UAV flight paths planned by the hybrid particle swarm optimization-Aquila optimization algorithm (HPSOCAOA) are shown in Figure 7. The starting position of the UAV is predefined, and the destination coordinates are listed in Table 7.

Figure 7. Specifically, (a1,b1) correspond to the side view and top view of Experiment 1, (a2,b2) represent the side view and top view of Experiment 2, (a3,b3) denote the side view and top view of Experiment 3, and (a4,b4) stand for the side view and top view of Experiment 4. All three drones in the figure can depart from the starting point, navigate around obstacles, and reach the destination, demonstrating the feasibility of HPSOCAOA for multi-drone path planning in complex environments.

Table 7. Starting and Ending Point Coordinate Information.

Experiment Number	Starting Point	Finish Line
1	(180, 90, 150)	(800, 700, 100)
2	(100, 100, 200)	(800, 800, 150)
3	(100, 60, 120)	(800, 600, 120)
4	(150, 160, 80)	(900, 800, 150)

7.2. Ablation Experiment

We selected the unimproved CAOA algorithm, the CAOA algorithm only adopting the adaptive energy strategy (CAOAEnergy), and the CAOA algorithm only using the hybrid learning strategy (CAOATransmit). Under the same environmental conditions, parameter settings, and software environment, four simulation experiments were conducted for both single-UAV and multi-UAV scenarios, and the experimental results were compared and analyzed with the HPSOCAOA algorithm proposed in this paper. A total of 30 experiments were performed, and one of the results was selected for visualization analysis. The fitness curves are shown in Figure 8 and Figure 9.

Figure 8. Single-UAV experiments. (1) Fitness curves of Group 1 single-UAV experiments; (2) Fitness curves of Group 2 single-UAV experiments; (3) Fitness curves of Group 3 single-UAV experiments; (4) Fitness curves of Group 4 single-UAV experiments.

Figure 9. Multi-UAV experiments. (1) Fitness curves of Group 1 multi-UAV experiments; (2) Fitness curves of Group 2 multi-UAV experiments; (3) Fitness curves of Group 3 multi-UAV experiments; (4) Fitness curves of Group 4 multi-UAV experiments.

7.3. Results Analysis

As shown in Figure 8 and Figure 9, all four algorithms can be applied to UAV trajectory planning. In the comparative experiments conducted under the same simulation environment, the HPSOCAOA algorithm exhibits a rapid downward trend at the early stage of iteration, and its convergence speed is significantly faster than that of other algorithms. In terms of convergence accuracy, the experimental results of the 4 algorithms in single-UAV and multi-UAV scenarios are presented in Table 8.

Table 8. UAV ablation experimental results.

	Index	HPSOCAOA	CAOA	CAOA Energy	CAOA Transmit
Single UAV	Mean	429.78	539.63	593.26	536.05
	Std	30.12	23.13	16.47	61.18
	Min	395.99	516.64	571.78	497.13
Multi UAV	Mean	395.67	579.93	522.30	459.29
	Std	43.80	42.76	27.90	53.25
	Min	354.95	522.77	493.32	399.18

It is worth noting that the cross-learning mechanism relies on the PSO-CAOA collaborative architecture and cannot operate independently. Constrained by this framework structure, it is difficult to separately isolate and quantify the individual contribution of the population division mechanism, which constitutes an inherent limitation of the present ablation experiments.

As shown in Table 8, the HPSOCAOA algorithm exhibits excellent comprehensive performance in three key indicators: mean value, optimal value, and standard deviation under both single-UAV and multi-UAV scenarios.

In the single-UAV scenario, the mean value of the HPSOCAOA algorithm is 429.78, which is 20.36% lower than the CAOA algorithm (539.63), 27.56% lower than the CAOAEnergy algorithm (593.26), and 19.82% lower than the CAOATransmit algorithm (536.05). The optimal value of HPSOCAOA is 395.99, which is 23.35% better than CAOA, 30.74% better than CAOAEnergy, and 20.34% better than CAOATransmit, while it is slightly lacking in the standard deviation index and exhibits relatively weaker stability compared with several comparison algorithms.

In the multi-UAV scenario, the mean value of the HPSOCAOA algorithm reaches 395.67, which is 31.77% lower than the CAOA algorithm (579.93), 24.24% lower than the CAOAEnergy algorithm (522.30), and 13.09% lower than the CAOATransmit algorithm (459.29). The optimal value of HPSOCAOA is 354.95, which is 32.10% better than CAOA, 28.05% better than CAOAEnergy, and 11.08% better than CAOATransmit, with only a slight deficiency in the standard deviation indicator.

From the overall experimental results, both the adaptive energy strategy (CAOAEnergy) and the hybrid learning strategy (CAOATransmit) can effectively improve the performance of the original CAOA algorithm, and the hybrid learning strategy achieves a more prominent improvement in the overall performance of the algorithm. A comprehensive comparison in terms of mean value, optimal value, and stability shows that CAOATransmit possesses a more obvious optimization effect on the original CAOA algorithm than CAOAEnergy. It indicates that the hybrid learning strategy plays a more critical role in improving optimization accuracy, accelerating convergence speed, and enhancing operational stability. The HPSOCAOA algorithm, integrating adaptive energy strategy and hybrid learning strategy, achieves optimal results in all indicators, which verifies that there is a positive synergistic effect between the two improved strategies. Meanwhile, it further confirms that the hybrid learning strategy is the core factor to enhance the performance of the original CAOA algorithm. Furthermore, the convergence curve of HPSOCAOA shows a rapid decline in the early stages of iteration, a steady and stable decline in the middle stages, and early convergence to the global optimum in the later stages. This indicates that it achieves an excellent balance between global search and local optimization, can efficiently find the optimal solution, possesses high robustness, and demonstrates significant search advantages in complex mountainous environments.

7.4. Simulation Comparison Experiment

The original CAOA algorithm, the WOA algorithm, the PSO algorithm, and two newly proposed swarm intelligence algorithms from recent years, including the DBO algorithm and SWO algorithm, are selected. Under the same environmental conditions, parameter settings, and software environment, simulation comparison experiments are carried out for both single-UAV and multi-UAV scenarios, and the experimental results are compared and analyzed with the HPSOCAOA algorithm proposed in this paper. The convergence curves are shown in Figure 10.

Figure 10. Comparative experiments.

A quantitative analysis is conducted on the simulation results of single-UAV and multi-UAV scenarios. In the single-UAV scenario, the optimal fitness values of each algorithm are 362.27 for HPSOCAOA, 570.08 for CAOA, 604.13 for WOA, 505.15 for PSO, 413.46 for DBO, and 415.46 for SWO. Compared with CAOA, WOA, PSO, DBO, and SWO, the fitness values of HPSOCAOA are reduced by 36.45%, 40.03%, 28.28%, 12.38%, and 12.80%, respectively. In the multi-UAV scenario, the optimal fitness values are 359.68 for HPSOCAOA, 526.10 for CAOA, 702.10 for WOA, 528.89 for PSO, 505.56 for DBO, and 389.50 for SWO, with the corresponding fitness values reduced by 31.63%, 48.77%, 31.99%, 28.86%, and 7.66%. The experimental trends under the two scenarios are consistent, and the optimal fitness value of HPSOCAOA is lower than that of the original CAOA, traditional WOA, and PSO, as well as the novel swarm intelligence algorithms DBO and SWO. Compared with the original CAOA algorithm, the optimization ranges of fitness values reach 31.63% and 36.45%. Compared with traditional algorithms such as WOA and PSO, the maximum optimization improvement reaches 48.77%. Even when compared with the recently proposed DBO and SWO algorithms, HPSOCAOA still exhibits optimization advantages of 7.66% and 28.86%. The results fully demonstrate that the improved mechanism integrating adaptive energy and hybrid learning strategies can effectively compensate for the insufficient optimization accuracy of the original CAOA algorithm and significantly enhance the global search ability. The proposed HPSOCAOA presents obvious superiority in both single-UAV and multi-UAV path-planning scenarios, which also verifies the rationality and effectiveness of the improved strategies in this paper.

8. Conclusions

This paper proposes a UAV trajectory planning method based on the Hybrid Particle Swarm-Crocodile Ambush Optimization Algorithm (HPSOCAOA) by constructing a collaborative search architecture that integrates CAOA and PSO; designing an adaptive energy-consumption coefficient to dynamically balance the exploration and exploitation capabilities throughout the iterative process; and introducing cross-learning to transmit high-quality information. The algorithm demonstrates significant effectiveness in solving trajectory planning problems in mountainous 3D environments, leading to the following conclusions:

The application of the Hybrid Particle Swarm-Crocodile Ambush Optimization Algorithm (HPSOCAOA) to UAV trajectory planning verifies its feasibility in generating safe, smooth, and optimal paths in a three-dimensional mountainous environment, thereby achieving trajectory planning from the starting point to the destination.

Various improvement strategies effectively enhanced the algorithm’s ability to balance local optimization and global exploration, effectively addressing the shortcomings of single-algorithm approaches. This successfully overcame the premature convergence issue common in traditional algorithms during flight path planning, enabling the generation of more cost-effective flight paths for UAVs and significantly improving the algorithm’s optimization capability and stability.

In terms of convergence performance, HPSOCAOA demonstrates significantly higher efficiency than the comparison algorithms. Under identical simulation environments, parameters, and software conditions, in single-UAV and multi-UAV scenarios, the comparative experimental results show that the HPSOCAOA algorithm is significantly superior to the original CAOA algorithm, traditional WOA and PSO algorithms, as well as the novel swarm intelligence algorithms DBO and SWO in terms of optimal fitness value, mean value, and stability. The maximum optimization accuracy of the algorithm can be improved by 48.77%, exhibiting remarkable advantages in convergence performance and operational stability. The ablation experimental results indicate that both the adaptive energy strategy and the hybrid learning strategy can effectively enhance the optimization ability of the basic CAOA algorithm, among which the hybrid learning strategy plays a more critical role in improving the comprehensive performance of the algorithm. The integration of the improved strategies produces a positive synergistic effect, enabling the HPSOCAOA algorithm to achieve optimal comprehensive performance in both single-UAV and multi-UAV path-planning tasks, which verifies the effectiveness and rationality of the improved mechanism proposed in this paper.

Overall, the proposed algorithm exhibits superior performance, effectiveness, and practicality in UAV trajectory planning. In future work, more practical constraints will be introduced, and additional improvement strategies will be adopted to further enhance the adaptability and robustness of the algorithm in real application scenarios.