Defining appropriate criteria tailored to specific operational scenarios is crucial for a UAV performing a mission. In this context, the objective is to minimize distance and utilize the flight path Xi=(Pi1,Pi2,…,Pij,…,Pin), where n denotes the number of waypoints the UAV must visit. Each waypoint aligns with a path node on the map, with the path nodes’ coordinates being Pij=(xij,yij,zij), and for two path nodes, the Euclidean distance is ‖PijPi,j+1→‖, resulting in the cost for the path length: (1)F1(Xi)=∑j=1n−1‖PijPi,j+1→‖

UAV path planning: Completing the path planning requires ensuring the safe movement of the UAV by mitigating threats posed by obstacles. Let K denote the set of all threats, with each threat modeled as a cylinder. According to the definition illustrated in Fig. 1, Ck represents the projected center coordinate of the k obstacle on the horizontal plane, with a corresponding radius Rk. While prior studies often approximate the UAV as a dimensionless particle due to its compact size, this work explicitly incorporates the UAV’s spatial occupancy. To enhance threat assessment precision. Specifically, the UAV is assigned a safety diameter D. For a path segment PijPi,j+1→, let dk denote the Euclidean distance from the obstacle center Ck to this segment. The associated threat cost is proportional to this distance dk. By integrating these geometric parameters, the proposed framework enables a more rigorous quantification of collision risks during UAV navigation through threat-laden environments.

The threat cost F2, a measure of how close the UAV is to obstacles, is computed by evaluating the distance from each waypoint Pij to the collision zone within the obstacle set K, relative to the safety margin S. (2){F2(Xi)=∑j=1n−1∑k=1KTk(PijPi,j+1→),Tk(PijPi,j+1→)={0,if dk>S+D+Rk(S+D+Rk)−dk,if D+Rk<dk≤S+D+Rk∞,if dk≤D+Rk

The safety diameter D is determined by the physical dimensions of the UAV, while the safety margin S is influenced by various operational factors, including environmental conditions and positioning system accuracy. For example, in static environments with reliable GPS, S typically ranges from several tens of meters.

During mission execution, the UAV’s flying altitude is typically constrained between predefined minimum and maximum thresholds, denoted as hmin and hmax, respectively. These constraints arise from application-specific demands, such as ensuring adequate resolution and field of view for visual data acquisition in surveying or search tasks. The altitude cost associated with a waypoint Pij is computed as follows: (3)Hij={|hij−(hmax+hmin)2|,if hmin≤hij≤hmax∞,otherwise

The altitude hij denotes the UAV’s vertical distance relative to the ground, while Hij is determined by Eq. (3), which ensures compliance with a set mean altitude while penalizing deviations beyond the acceptable range. As a result, the altitude cost function is expressed as: (4)F3(Xi)=∑j=1nHij

Evaluating the smoothing cost requires calculating both the turning angle and the climbing angle. As illustrated in Fig. 2 and mathematically defined in Eq. (6), the turning angle φij is defined as the angle between the two projected path segments on the Oxy plane. Let k→ denote the unit vector along the z-axis. The projected vector can then be mathematically expressed as:

(5)Pij′Pi,j+1′→=k→×(PijPi,j+1→×k→)

Thus, the turning angle is determined by: (6)φij=arctan⁡(‖Pij′Pi,j+1′→×Pi,j+1′Pi,j+2′→‖Pij′Pi,j+1′→⋅Pi,j+1′Pi,j+2′→)

The climb angle, denoted as ψij, is the angle between the ascent trajectory of the UAV and the horizontal plane. Specifically, the climb angle ψij represents the angular deviation between the actual path segment PijPi,j+1→ and its horizontal projection Pij′Pi,j+1′→. The term zi,j+1−zij corresponds to the difference in altitude (vertical displacement) between two consecutive waypoints. Thus, the climb angle is calculated as: (7)ψij=arctan⁡(zi,j+1−zij‖Pij′Pi,j+1′→‖)

The smoothing cost is formulated as follows: (8)F4(Xi)=a1∑j=1n−2φij+a2∑j=1n−1|ψij−ψi,j−1|where a1 and a2 are the penalty coefficients for the turning angle and climbing angle, respectively.

Taking into account the optimality, safety, and feasibility constraints related to the path Xi, the overall cost function for the entire path Xi can be defined through Eqs. (1) to (8): (9)F(Xi)=∑k=14bkFk(Xi)

Choosing the most suitable function for UAV path planning is crucial. This study uses the optimal overall cost function in complex environments with multiple threats during UAV navigation as a suitable function. The coefficients bk represent the weights assigned to each cost component.

The SPSO algorithm encodes each flight path as a sequence of vectors, where each vector represents the UAV’s movement from one waypoint to the next [14]. These vectors are expressed in spherical coordinates and comprise three components: magnitude ρ∈(0,path length), elevation angle ψ∈(−π2,π2), and azimuth angle φ∈(−π,π). Consequently, a flight path Ωi consisting of n waypoints are represented as a spherical vector sequence with N=n−2 elements: (10)Ωi =(ρi1,ψi1,ϕi1,ρi2,ψi2,ϕi2,⋯,ρiN,ψiN,ϕiN),N=n−2

The spherical vector (ρij,ψij,ϕij) is denoted as uij. For a swarm of M particles searching in an N-dimensional space (where N corresponds to the number of spherical vector components representing the path), the update equations of SPSO are defined as follows: (11)Δuijt+1←wt⋅Δuijt+cgpi(t)r1jt(qijt−uijt)+cggi(t)r2jt(qgjt−uijt) (12)uijt+1←uijt+Δuijt+1, (i=1,2,…,M;j=1,2,…,N)

In these equations, cgpi(t) and cggi(t) represent the cognitive and social acceleration coefficients at iteration t, respectively, whose adaptive design is detailed in Section 3.2. The terms r1jt and r2jt are uniform random numbers within [0,1] sampled for each dimension j and iteration t, introducing stochasticity to the search process. The local best and global best positions of particles are represented by the vector sets Qi=(qi1,qi2,…,qiN) and Gi=(gi1,gi2,…,giN), respectively. Determining Qi and Gi requires constructing the vector flight path map Ωi, which directly evaluates the relevant costs. The transformation from the spherical coordinate vector uij=(ρij,ψij,ϕij)∈ Ωi to the Cartesian waypoint Pij=(xij,yij,zij)∈Xi is performed as follows: (13)xij=xi,j−1+ρijsin⁡ψijcos⁡ϕij (14)yij=yi,j−1+ρijsin⁡ψijsin⁡ϕij (15)zij=zi,j−1+ρijcos⁡ψij

Using spherical vectors in SPSO mainly relies on improvement in navigation safety through establishing mathematical correlations between the vector components (magnitude, elevation, and azimuth) and the UAV’s kinematic parameters (velocity, turning angle, and climb angle). With this, particle-based solutions search through space-configuration instead of spaceCartesian, thereby increasing the possibility of finding good quality trajectories. Notably, the elevation and azimuth parameters inherently enforce constraints on steering and climb angles through their geometric definitions, substantially constraining the solution space.

Conventional UAV path planning employs metaheuristic algorithms with static parameters to strike a balance between exploration and exploitation. Fixed acceleration coefficients and inertia weights lack adaptability to nonlinear cost functions or dynamic threats. Similarly, standard SPSO uses static spherical vector encoding with constant social-cognitive coefficients, integrating UAV kinematics geometrically but missing the dynamic exploration-exploitation balance. This rigidity causes slow convergence and local optima trapping in complex environments due to unresponsiveness to changing solution landscapes. To address this limitation, a dual Sigmoid-based adaptive acceleration mechanism modulating cognitive and social coefficients in real-time via particle-best Euclidean distances and iteration phases. The proposed mechanism for dynamic search landscapes boosts optimization efficiency and solution quality.

The proposed mechanism adaptively adjusts the cognitive coefficient cgpi(t) and the social coefficient cggi(t) for each particle i at iteration t. The adjustment is governed by the real-time Euclidean distances between the particle’s current position xi(t), and its personal best position pi(t), as well as the global best position gi(t) found by the entire swarm so far. The coefficients are defined by the following Sigmoid functions, where b1 and b2 are constant scaling parameters that define the maximum possible values for the cognitive and social coefficients, respectively. They are determined empirically to set an upper bound for the acceleration influence. The terms ‖pi(t)−xi(t)‖ and ‖gi(t)−xi(t)‖ denote the Euclidean distances representing how far the particle is from its personal best solution and global best solution, respectively. (16)cgpi(t)=b11+e−a1(t)⋅‖pi(t)−xi(t)‖ (17)cggi(t)=b21+e−a2(t)⋅‖gi(t)−xi(t)‖

The parameters a1(t) and a2(t) are time-varying steepness parameters that control the sensitivity of the Sigmoid functions to these distances. Their values evolve throughout the optimization process to enforce distinct behaviors during different search phases. The parameter a1(t), associated with the cognitive component, increases linearly with the iteration count t. This progressive increase amplifies the sensitivity of cgpi(t) to the distance ‖pi(t)−xi(t)‖ during the later stages of optimization. As a result, particles are increasingly encouraged to refine their search around their personal best positions, enhancing local exploitation. Conversely, a2(t), governing the social component, decreases linearly with t. This reduction boosts the responsiveness of cggi(t) to the global distance ‖gi(t)−xi(t)‖ primarily during the early phases, promoting robust global exploration by drawing particles more strongly towards the swarm’s best-found region. The opposing evolutionary trends of a1(t) and a2(t) ensure a smooth and automatic transition from a broad, exploratory search to a focused, exploitative refinement. Notably, the sum a1(t)+a2(t)=3ainitial remains constant throughout the process, ensuring computational balance. (18)a1(t)=ainitial⋅(1+ttmax) (19)a2(t)=ainitial⋅(2−ttmax)

The inherent monotonic increasing property of the Sigmoid function is central to the mechanism’s behavior. The acceleration coefficients increase as the corresponding distances increase. When a particle is far from a the best position, the argument of the Sigmoid becomes a large positive number, causing cgpi(t) and cggi(t) to approach their maximum values b1 and b2. This generates strong acceleration forces, compelling the particle to move rapidly towards the promising best position to bridge the large gap. Conversely, when a particle is already close to a best position, the argument of the Sigmoid becomes small, causing the coefficients to diminish towards lower values. This allows for gentle, precise adjustments in the vicinity of the best position, preventing overshooting and facilitating fine-tuned local search. This intelligent design aligns the magnitude of velocity updates with the immediate optimization need: aggressive movement towards distant promising areas and cautious refinement within nearby regions. The intelligent design ensures vigorous exploration of vast search spaces while maintaining stable exploitation in promising regions without disrupting the convergence process.

By incorporating this dynamic state awareness and phase-dependent tuning into the acceleration coefficients, the proposed mechanism effectively addresses the rigidity of static PSO parameter configurations. The proposed mechanism offers a more adaptive, efficient, and intelligent approach to balancing exploration and exploitation in complex UAV path planning problems.

To overcome the limitations of the SPSO algorithm in balancing global exploration and local exploitation during UAV path planning, which often leads to premature convergence and slow optimization, this paper incorporates a lens imaging-based opposition learning strategy [23]. This method enhances the algorithm’s ability to explore uncharted regions of the search space while simultaneously refining solutions in promising areas, thereby improving both population diversity and convergence speed. The core idea is inspired by the conjugate relationship between an object and its image in geometric optics, where a convex lens generates a symmetrical image of an object. By analogizing the current solution (particle position) to an object, its opposition solution (mirror point) can be dynamically generated within the solution space, facilitating a bidirectional search mechanism.

Mathematically, for a particle located at position x within the current search boundaries [a,b] in a given dimension, its lens-based opposition solution x∗ is calculated using the lens imaging formula: (20)uij∗=a+b2+a+b2k−uijk

Here, k>0 is a scaling factor that controls the degree of opposition. This factor is dynamically adjusted throughout the iterations to balance exploration and exploitation. Initially, a larger k value pushes the opposition solution closer to the center of the search space, encouraging exploration of broader regions. As the optimization progresses, k is linearly decreased to a smaller value, which draws the opposition solution nearer to the current solution, thus promoting local refinement around promising areas. The dynamic update of k is governed by: (21)kt=kmax−kmax−kmintmax⋅twhere t is the current iteration and tmax is the maximum number of iterations. Furthermore, the search boundaries [a,b] for each dimension are not fixed but are dynamically contracted based on the distribution of the current population, focusing the search on increasingly promising regions. The process of generating an opposition solution via this strategy is conceptually illustrated in Fig. 3, which depicts the optical analogy of an object at x forming an image at x∗ through a lens placed at the midpoint o=(a+b)/2.

The integration of this lens-based opposition learning into the SPSO framework occurs after the standard velocity and position update. For each particle, an opposition solution is generated in the spherical coordinate space using Eq. (20). The fitness of this new solution is evaluated and compared against the original particle’s fitness. If the opposition solution yields a better fitness value, the opposition solution replaces the current particle in the swarm. This mechanism injects new, high-quality solutions into the population, enhancing diversity and providing an effective means to escape local optima. The strategy systematically improves the algorithm’s performance by leveraging the inherent duality and symmetry of optical systems, resulting in a more robust and efficient balance between exploration and exploitation for complex UAV path planning problems.

The implementation of the DW-SPSO algorithm in the trajectory planning of UAVs incorporates spherical vector encoding, dynamic weight adjustment, and a lens-based opposition learning strategy, thereby establishing a comprehensive optimization framework. The algorithm’s core task is to balance global exploration and local exploitation, enhancing solution diversity through concepts from geometric optics theory. This is an attempt to create relatively safe and smooth flight paths in dangerous threat environments. Flowchart of DW-SPSO, as delineated in Algorithm 1, shares a structural resemblance with other PSO algorithms, including parameter initialization, particle generation, and swarm evolution. Nevertheless, DW-SPSO primarily distinguishes itself from other PSO algorithms through the representation of particle positions and velocities, as well as the update equations. This unique design enables DW-SPSO to address the path planning issue for UAVs in complex environments more effectively, ensuring their safe and efficient task execution.

The evaluation scenarios are based on the public 5 metre DEM of Australia, derived from a LiDAR dataset, openly provided by Australia [24]. This dataset is available under a Creative Commons Attribution 4.0 International Licence from the official portal (https://elevation.fsdf.org.au/). Two distinct terrain patches on Christmas Island (approximate extent: 105.53∘E to 105.65∘E, 10.42∘S to 10.55∘S) were selected and extended to create two benchmarking scenarios: the ChristmasTerrain model and the TerrainData model. The number and placement of threats, represented by green cylinders, vary in complexity between these scenarios. Table 1 displays the environmental parameters for the simple and complex obstacle configurations within the ChristmasTerrain model, while Table 2 provides the corresponding parameters for the TerrainData model.

The proposed DW-SPSO is compared with standard PSO [25] and some state-of-the-art PSO variant algorithms, including APSO [26], AWPSO [27], DSPSO [28], SPSO [14], VPPSO [29], PPSOBAS [30] and TS-CEPSO [31]. APSO adapts parameters based on the population state. AWPSO employs an S-type function-based weighting strategy. DSPSO incorporates dynamic differential mutation. SPSO utilizes spherical coordinates for path encoding. VPPSO uses a “velocity pausing” mechanism to maintain diversity. PPSOBAS hybridizes PSO with the Beetle Antennae Search algorithm. TS-CEPSO integrates chaotic maps and feasibility rules for complex constraints.

To ensure experimental fairness and comparability, all algorithms were configured under consistent conditions: a population size of 100 and a maximum of 100 iterations. The parameter settings for all compared algorithms were meticulously adopted from their respective original publications or authoritative implementations to ensure a faithful and unbiased comparison. The complete parameter configurations are summarized in Table 3.

For the proposed DW-SPSO algorithm, the key parameters of the dual Sigmoid mechanism (ainitial, b1, b2) were rigorously optimized. An initial sensitivity analysis of over 200 candidate parameter sets identified these five parameters as the most influential. Bayesian optimization was then employed to minimize the average fitness (Eq. 9) within the defined search spaces (ainitial∈[10−6,10−3]; b1,b2∈[0.1,2.0]), converging to the optimal combination listed in Table 3. The algorithm demonstrated robustness to ±10% variations in these key parameters, with performance degradation remaining below 5%, confirming the stability of the selected configuration.

A paired sample t-test [32] with a significance level of α=0.05 (95% confidence) is used to assess the statistical significance of the performance differences between DW-SPSO and all other algorithms. The notation D+ indicates that the performance of DW-SPSO is statistically better than the compared algorithm, D− indicates the DW-SPSO performance is statistically worse, and N indicates no statistically significant difference (Not Applicable). NA means Not Applicable.

To comprehensively assess the performance of the proposed DW-SPSO algorithm. The experiments were performed on two distinct terrain models (ChristmasTerrain and TerrainData) with varying obstacle complexities. They examined the result in terms of the quality of the path, whether it converged or not, as well as in terms of significance.

To further quantify the path quality, a detailed analysis was conducted on three critical metrics derived from Eqs. (6)–(8): climb angle, safe distance, and smoothness score. As shown in Table 4, DW-SPSO achieves the best overall performance with a balanced combination of these metrics. Specifically, the algorithm maintains a moderate climb angle of 5.30∘, which is significantly lower than most competitors (e.g., APSO: 8.90∘, SPSO: 8.49∘), indicating more stable altitude transitions. In terms of safety, DW-SPSO maintains a safe distance of 4.98 m, comparable to the best-performing algorithms, while ensuring collision avoidance. Most notably, DW-SPSO achieves the highest smoothness score of 85.02, demonstrating its superior ability to generate paths with minimal abrupt turns and altitude changes. This balanced performance across all three metrics validates the effectiveness of the spherical vector encoding and dynamic weight adjustment in producing practically feasible UAV paths.

Fig. 4 shows the top-down views of the paths taken by the compared PSO algorithms for the Christmas Terrain model and TerrainData model, respectively, All algorithms successfully generated feasible paths that satisfied constraints like path length, obstacle avoidance, and smoothness. Notably, the paths produced by DW-SPSO exhibit superior adaptability to complex environments, with smoother transitions and fewer sharp turns, as highlighted in Fig. 5. This demonstrates the effectiveness of the dynamic weight adjustment mechanism and lens-based opposition learning in enhancing path quality.

DW-SPSO’s computational efficiency was evaluated against standard PSO and SPSO. All three algorithms share theoretical time complexity O(M⋅N⋅T) for population size M, dimensionality N, and iterations T. Empirically, DW-SPSO incurs 15%–20% extra time per iteration due to dual Sigmoid adaptation and lens opposition learning. This overhead is justified by superior convergence (Fig. 6), achieving better solutions in fewer iterations. Scalability tests (50–500 particles) confirm linear scaling, ensuring practical UAV applications. The algorithm’s parallel fitness evaluation suits hardware acceleration on embedded GPUs (e.g., NVIDIA Jetson), where convergence gains offset iteration overhead for real-time replanning.

Fig. 6 shows the convergence trend of the best fitness values with respect to iteration. DW-SPSO is always faster and achieves better fitness values compared to other algorithms, indicating its rapid convergence and robust optimization capability. The dual Sigmoid-based adaptive acceleration mechanism enables DW-SPSO to balance exploration and exploitation effectively, avoiding premature convergence to local optima.

Table 5 presents a summary of the average fitness values. All variants converge similarly, with the exception of VPPSO, due to its velocity pausing mechanism. This algorithm deliberately halts velocity updates when diversity metrics fall below certain thresholds, resulting in periodic plateaus. Additionally, standard deviations and paired t-test results are provided for the ChristmasTerrain and TerrainData models, respectively. DW-SPSO demonstrates superior performance compared to most algorithms, as indicated by its lower average fitness values and smaller standard deviations, which suggest higher solution quality and stability. The t-test results (denoted as D+) confirm that the performance improvements of DW-SPSO are statistically significant in most scenarios.

The Wilcoxon signed-rank test [33] results unequivocally demonstrate the superiority of the proposed DW-SPSO algorithm. As shown in Table 6, DW-SPSO achieves statistically significant improvements (p<0.05) over all other state-of-the-art algorithms, including the baseline SPSO (p=0.03485). The p-values for comparisons with algorithms like APSO, AWPSO, and TS-CEPSO are exceedingly small (p<1e−50), providing overwhelming evidence that the performance differences are not due to chance. This comprehensive statistical superiority validates the effectiveness of the dynamic weight adjustment and lens-based opposition learning strategies introduced in this work.

To provide a more robust statistical verification that is insensitive to non-normal data distributions, the Wilcoxon signed-rank test was further conducted the Wilcoxon signed-rank test on the results from 30 independent runs. The detailed p-values are comprehensively summarized in Table 6. The analysis reveals that DW-SPSO achieves statistically significant improvements (p<0.05) over all other state-of-the-art algorithms across the majority of scenarios. For instance, when compared to the baseline SPSO, the superiority of DW-SPSO is statistically significant (p=0.03485). Notably, the p-values for comparisons against algorithms such as APSO, AWPSO, and TS-CEPSO are exceedingly small (p<1e−50), providing overwhelming evidence that the observed performance enhancements are not attributable to random chance. This rigorous non-parametric analysis, coupled with the earlier t-test results, offers comprehensive and robust statistical evidence that firmly validates the effectiveness of the proposed dynamic weight adjustment mechanism and lens-based opposition learning strategy.

To thoroughly analyze the contributions of individual components in the DW-SPSO algorithm and evaluate parameter sensitivity, systematic ablation experiments and sensitivity tests. The tests were performed in a simplified 150 × 150 environment with 6 waypoints, resulting in an 18-dimensional search space.

Ablation studies were conducted by systematically disabling key components of the proposed algorithm. As shown in Table 7 (with the best-performing configuration highlighted in bold), the complete DW-SPSO achieved the best performance with an average fitness of 2520.86. Removing the dual Sigmoid weighting mechanism resulted in a 13.9% performance degradation (fitness: 2871.76), demonstrating its crucial role in balancing exploration and exploitation. Disabling the lens opposition learning caused a 1.9% performance drop (fitness: 2569.83), confirming its contribution to solution diversity. The baseline PSO without any enhancements performed worse (fitness: 3296.40), highlighting the collective importance of both proposed mechanisms.

Parameter sensitivity analysis was extended beyond the typical ±10% range to assess robustness under more extreme conditions (±60% variation). The social coefficient scaling factor b2 proved most sensitive, with performance deviations up to 14.1%. The initial steepness parameter ainitial and distance scaling factor d2 showed moderate sensitivity (5.7% deviation), while b1 and d1 demonstrated better stability with deviations below 6%. Notably, the algorithm maintained reasonable performance even under these extreme parameter variations, confirming its robustness for practical applications.

To address the practical deployment of the proposed DW-SPSO algorithm, this section discusses its computational characteristics and potential integration with UAV hardware platforms. The algorithm’s inherent parallelism in fitness evaluation makes it suitable for hardware acceleration. Recent studies have demonstrated that the successful implementation of PSO variants on embedded systems is commonly used in UAV autopilots. For instance, El-Metwally et al. implemented a smart decision-making framework on NVIDIA Jetson platforms, showing real-time capability for autonomous navigation tasks. Similarly, research by Alhusseini et al. [20] showcased adaptive PSO optimization on embedded GPU architectures, achieving significant speedup for complex optimization problems.

While current simulations validate DW-SPSO for offline planning, its computational structure shows strong real-time replanning potential. Spherical vector encoding reduces dimensionality vs. Cartesian representations, and adaptive mechanisms minimize redundant computations. Future work will implement DW-SPSO on embedded processors (NVIDIA Jetson Orin, Qualcomm Snapdragon Ride), leveraging parallel computing for dynamic environment performance.