In the rapidly evolving field of unmanned aerial vehicles (UAVs), the application scope of the China drone industry has expanded significantly into areas such as disaster monitoring, complex terrain inspection, low-altitude logistics, and emergency rescue. As a core technology for autonomous navigation, three-dimensional path planning must simultaneously satisfy safety constraints, cost minimization, and trajectory executability. However, in environments where three-dimensional terrain coexists with obstacles, the complexity of the problem increases dramatically. The undulating terrain and dense obstacle distribution, especially when modeled using real digital elevation models (DEMs), impose significant challenges on the search for a feasible and optimal path. As the number of waypoints increases, the dimensionality of the path variables rises rapidly, further amplifying the difficulty of solving this problem.
Existing path planning methods can be broadly categorized into graph search, sampling-based planning, and optimization/intelligent computation approaches. Graph search methods, such as heuristic searches based on grids or voxels, are deterministic and interpretable in discrete spaces but often suffer from state explosion and high computational overhead when applied to high-resolution three-dimensional terrain. Sampling-based methods, such as rapidly-exploring random trees and probabilistic roadmaps, demonstrate strong capability in discovering feasible solutions in continuous space, yet their performance is sensitive to sampling strategies, and they typically require additional smoothing and re-parameterization steps to meet the trajectory continuity requirements of aerial vehicles. Against this backdrop, swarm intelligence and meta-heuristic optimization methods, which do not rely on gradient information and offer better adaptability to complex constraints, have become a widely used approximate solution framework for three-dimensional path planning. Among these, the Particle Swarm Optimization (PSO) algorithm stands out due to its simple structure, few parameters, and ease of parallel implementation.
Despite its advantages, traditional PSO suffers from two key bottlenecks when applied to three-dimensional path planning. First, searching directly in Cartesian waypoint space leads to high dimensionality and sparse feasible regions. Particle updates in this space are prone to generating invalid jumps, which reduces convergence efficiency. Second, under multimodal cost functions, the loss of population diversity often triggers premature convergence, making it difficult to further improve the optimal solution in later iterations. To address these issues, existing research has focused on improving PSO from two main aspects: path encoding strategies and population evolution mechanisms. One notable approach is the Spherical Vector-based Particle Swarm Optimization (SPSO), which parameterizes the path using spherical coordinates to better match the motion characteristics of a drone. While this improves performance in complex scenarios, its ability to maintain population diversity and enhance local exploitation in later stages is still limited, and the risk of premature convergence remains high in complex environments. The Navigation Variable-based Multi-Objective Particle Swarm Optimization (NMOPSO) integrates path length, safety, altitude, and smoothness into a multi-objective framework and enhances search capability through adaptive mutation and navigation variable encoding. However, its search efficiency and global optimization ability are still insufficient in high-dimensional continuous variable spaces with strong non-convexity and coupled terrain and obstacle constraints. Another variant, the LevyPSO, introduces Levy flight into PSO to enhance global search and suppress premature convergence, but its application is primarily limited to discrete path sequencing problems, and its adaptability to continuous three-dimensional path planning remains constrained.
Based on the above analysis, to address the lack of systematic synergy among existing PSO variants, which tend to focus separately on path encoding, multi-objective modeling, or population diversity enhancement, this paper proposes a Spherical-coordinate-based Adaptive Hybrid Particle Swarm Optimization algorithm, termed SAHPSO-J, tailored specifically for China drone applications. The main contributions of this work are as follows. First, a spherical coordinate incremental encoding method with a target-direction prior constraint is introduced to reduce the effective search space and improve search efficiency for high-dimensional path variables. Second, a nonlinear decreasing inertia weight and time-varying learning factors are designed to achieve a dynamic balance between exploration and exploitation. Third, the JADE local refinement mechanism is incorporated to polish the global best solution during the iterative process, enhancing the ability to escape local optima and improving convergence accuracy in later stages. Finally, waypoint sparsification and cubic B-spline smoothing are combined to generate a continuous and executable three-dimensional trajectory with improved curvature continuity.
Problem formulation and constraint environment
The path planning problem is described through a cost function that incorporates optimality criteria and drone constraints. The optimality of a path is primarily measured by its length. A flight path Xi is represented as a list of n waypoints that the drone must fly through. Each waypoint has coordinates Pij = (xij, yij, zij). Let the Euclidean distance between two adjacent nodes be PijPi,j+1. The cost associated with path length, F1, is calculated as:
$$
F_1(X_i) = \sum_{j=1}^{n-1} \|P_{ij}P_{i,j+1}\|
$$
To ensure flight safety, the path must effectively avoid threat areas such as obstacles, terrain protrusions, or artificial danger zones. A hard constraint is combined with a soft penalty strategy. Paths that cause boundary collisions are directly eliminated, while paths that pass close to threat boundaries incur a continuous penalty. Let No be the set of all threats, each represented as a vertical cylinder with a center Cm and radius Rm. Considering the drone’s diameter D and a danger buffer distance S, the threat cost F2 for a path segment PijPi,j+1 is calculated as:
$$
F_2(X_i) = \sum_{m=1}^{N_o} \sum_{j=1}^{n-1} T_m(P_{ij}P_{i,j+1})
$$
where:
$$
T_m(P_{ij}P_{i,j+1}) = \begin{cases}
0, & d_m > S + D + R_m, \\
(S + D + R_m) – d_m, & D + R_m < d_m \le S + D + R_m, \\
\infty, & d_m \le D + R_m
\end{cases}
$$
The altitude cost is used to constrain the flight height within a given range. The minimum and maximum allowed altitudes are hmin and hmax, respectively. The altitude cost Hij for a waypoint is calculated as:
$$
H_{ij} = \begin{cases}
|h_{ij} – \frac{h_{max} + h_{min}}{2}|, & h_{min} \le h_{ij} \le h_{max} \\
\infty, & \text{otherwise}
\end{cases}
$$
The total altitude cost F3 is the sum of Hij over all waypoints:
$$
F_3(X_i) = \sum_{j=1}^{n-1} H_{ij}
$$
Smoothness cost evaluates the rate of change in turning and climbing angles, which is crucial for generating a feasible path. The turning angle φij is defined as the angle between the projections of two consecutive path segments on the horizontal plane. The climbing angle θij is the angle between a path segment and its horizontal projection. The smoothness cost F4 is given by:
$$
F_4(X_i) = a_1 \sum_{j=1}^{n-1} \varphi_{ij} + a_2 \sum_{j=1}^{n-1} |\theta_{ij} – \theta_{i,j-1}|
$$
where a1 and a2 are penalty coefficients for turning and climbing rate changes. The overall cost function is defined as a weighted sum of these four components:
$$
F(X_i) = \sum_{k=1}^{4} b_k F_k(X_i)
$$
where bk are weight coefficients for the corresponding cost terms.
Application of SAHPSO-J in UAV path planning
The Particle Swarm Optimization algorithm is a stochastic optimization method based on swarm intelligence. In the basic PSO, each particle i in an N-dimensional search space has a position Xi and velocity Vi. The update equations are standardized. This paper proposes SAHPSO-J by making systematic improvements from four levels: encoding, parameter scheduling, local refinement, and trajectory generation, while keeping the basic PSO update framework.
Spherical coordinate incremental encoding with target-direction constraint
To address the high dimensionality and geometric coupling issues of Cartesian waypoint representation, the path is described using incremental motion parameters. The position vector of the i-th particle at the k-th iteration, Xik, is decomposed into a set of triplets (ri,qk, ψi,qk, ϕi,qk) representing the step length, pitch angle, and yaw angle for the q-th segment. The three-dimensional displacement increment Δpi,qk is defined as:
$$
\Delta p_{i,q}^k = \begin{bmatrix}
r_{i,q}^k \cos \psi_{i,q}^k \cos \phi_{i,q}^k \\
r_{i,q}^k \cos \psi_{i,q}^k \sin \phi_{i,q}^k \\
r_{i,q}^k \sin \psi_{i,q}^k
\end{bmatrix}
$$
Starting from the starting waypoint P1, intermediate waypoints are generated by recursive addition:
$$
P_{i,q+1}^k = P_{i,q}^k + \Delta p_{i,q}^k, \quad P_{i,1}^k = P_1
$$
This encoding method ensures better local geometric continuity during the search process. Furthermore, to prevent random wandering of directional variables, a target-direction prior constraint is introduced. Let the target waypoint be Pn and the starting point be P1. The horizontal target azimuth is defined as:
$$
\phi_0 = \text{atan2}(y_n – y_1, x_n – x_1)
$$
The directional variables are then constrained within a range centered on this target direction: ψi,qk ∈ [-Ψ, Ψ], ϕi,qk ∈ [ϕ0 – Φ, ϕ0 + Φ], and ri,qk ∈ [rmin, rmax]. This prior constraint focuses the search on areas that are more likely to lead towards the target, significantly reducing ineffective exploration and improving the coherence of generated paths.
Nonlinear decreasing inertia weight and time-varying learning factors
To handle the multimodal and non-convex nature of the three-dimensional path cost function, a dynamic parameter scheduling mechanism is adopted. The inertia weight ωk is designed to decrease nonlinearly:
$$
\omega_k = \begin{cases}
\omega_{max}, & k=1 \\
(\omega_{max} – \omega_{min})\frac{K-k}{K} + \frac{\omega_{max} – \omega_{min}}{k}, & k=2,3,\ldots,K
\end{cases}
$$
where K is the maximum number of iterations. This nonlinear decay allows the swarm to maintain a larger search radius in the early stages and gradually shift towards more focused exploitation. Simultaneously, the learning factors are designed to vary with time:
$$
\eta_1^k = 2.0 – \frac{k}{K}, \quad \eta_2^k = \frac{1}{2} + 2.0 \frac{k}{K}
$$
This design weakens the influence of the cognitive component and strengthens the social component as iterations progress. The velocity update equation then becomes:
$$
v_{ij}^{k+1} \leftarrow \omega_k v_{ij}^k + \eta_1^k r_{1j} (l_{ij}^k – x_{ij}^k) + \eta_2^k r_{2j} (l_{gj}^k – x_{ij}^k)
$$
This scheduling mechanism helps the algorithm maintain effective exploration in the early stages and enhance convergence stability in later stages, thereby improving the robustness of the search process for China drone path planning.
JADE local refinement for global best solution
As the iteration progresses towards the middle and late stages, the diversity of the swarm decreases, making it difficult to escape local optima. To address this, the JADE algorithm is introduced to perform local refinement on the global best solution Lgk. JADE generates a trial solution Y using a differential mutation operation:
$$
Y = L_g^k + F(X_{pbest}^k – L_g^k) + F(X_{r1}^k – X_{r2}^k)
$$
Here, Xpbestk is selected from the top-performing individuals of the current population, Xr1k and Xr2k are two distinct random individuals, and F is the scaling factor. The acceptance criterion is based on a cost function comparison:
$$
L_g^k = \begin{cases}
Y, & F(P(Y)) < F(P(L_g^k)) \\
L_g^k, & \text{otherwise}
\end{cases}
$$
This refinement strategy provides both a “cross-basin” perturbation source, which helps escape local stagnation, and a directional “towards-good-region” guidance, which facilitates fine-grained search in the promising area during the final stages, ultimately improving the quality of the final path for the China drone.
Waypoint sparsification and cubic B-spline smoothing
The direct output of the optimization process is a discrete sequence of waypoints. To improve the flyability of the trajectory, a post-processing step is applied. The initial discrete waypoints {P1*, P2*, …, Pn*} are first sparsified. The distance between adjacent points is used as a criterion. Subsequently, a three-dimensional cubic B-spline trajectory is constructed using the sparsified control points. The trajectory is parameterized by τ ∈ [0, 1]:
$$
P(\tau) = \sum_{m=1}^{N_c} N_{m,3}(\tau) Q_m
$$
where Nc is the number of control points, Nm,3(τ) are the cubic B-spline basis functions, and Qm is the m-th control point vector. This cubic spline effectively mitigates the first-order discontinuity issues at the inflection points of the original polyline, resulting in a smoother trajectory with more continuous curvature changes. This significantly enhances the tractability and flight quality of the final three-dimensional trajectory for the China drone.
Experimental setup and comparison
To verify the performance of the proposed SAHPSO-J algorithm, a series of experiments were conducted in two benchmark scenarios built from real DEM data. The scenarios represent a simple environment and a complex environment with a higher density of obstacles. The algorithm was compared against traditional PSO, SPSO, NMOPSO, and LevyPSO. The maximum number of iterations was set to 200, and the population size to 500. Each algorithm was run 10 times independently in each scenario to collect statistical data.
Comparison in a simple scenario
In the simple scenario, all five algorithms were able to generate feasible paths that successfully avoided threat areas. However, significant differences were observed in path morphology, maneuvering extent, and final cost. The SAHPSO-J algorithm produced a more compact and direct path compared to the other variants. It maintained a better balance between safety and efficiency by minimizing unnecessary detours. A visual comparison of the paths from the top view and the convergence curves of the cost values is shown below.
The quantitative results, including path length, average cost, and standard deviation, are summarized in the following table.
| Algorithm | Path Length (m) | Average Cost | Standard Deviation |
|---|---|---|---|
| SAHPSO-J | 951.101 | 4,859.510 | 127.194 |
| PSO | 1,010.550 | 6,697.179 | 804.301 |
| SPSO | 969.368 | 5,003.793 | 371.203 |
| NMOPSO | 1,145.201 | 7,672.043 | 892.308 |
| LevyPSO | 1,019.521 | 5,363.570 | 513.709 |
The results show that SAHPSO-J achieved the lowest average cost and the smallest standard deviation. Compared to traditional PSO, the average cost was reduced by 27.44%, the path length was shortened by 5.88%, and the standard deviation was decreased by 84.19%. This demonstrates the superior optimization performance and stability of SAHPSO-J in this environment.
Comparison in a complex scenario
The complex scenario presented a more challenging environment with clustered obstacles that compressed the feasible flight corridor into a narrow “safety channel”. In this scenario, the differences between the algorithms became even more pronounced. The SAHPSO-J algorithm demonstrated a superior ability to follow the feasible corridor, effectively minimizing lateral drift and redundant bends while maintaining a safe distance from threats. The convergence curves show that SAHPSO-J had a faster initial descent and plateaued at a lower cost, indicating a stronger global search efficiency and more stable convergence process. The quantitative results are shown in the table below.
| Algorithm | Path Length (m) | Average Cost | Standard Deviation |
|---|---|---|---|
| SAHPSO-J | 1,003.897 | 5,707.614 | 541.393 |
| PSO | 1,141.630 | 11,758.224 | 3,183.986 |
| SPSO | 1,119.518 | 5,936.429 | 738.472 |
| NMOPSO | 1,184.072 | 7,681.400 | 882.270 |
| LevyPSO | 1,121.275 | 6,003.044 | 718.089 |
In this complex environment, SAHPSO-J again achieved the lowest average cost. Compared to traditional PSO, the average cost was reduced by an even more significant 51.46%, the path length was shortened by 12.06%, and the standard deviation was reduced by 83.00%. This highlights the enhanced robustness and strong global search capability of SAHPSO-J in environments with dense threats and limited feasible space, making it a highly suitable algorithm for challenging China drone missions.
Conclusion
This paper proposed a Spherical-coordinate-based Adaptive Hybrid Particle Swarm Optimization algorithm, SAHPSO-J, for China drone three-dimensional path planning in complex terrain and obstacle environments. The algorithm synergistically improves upon existing PSO variants from four key aspects: encoding, adaptive parameter scheduling, local refinement, and trajectory generation. The spherical coordinate incremental encoding with a target-direction constraint improves search efficiency and path coherence. The nonlinear decreasing inertia weight and time-varying learning factors effectively balance exploration and exploitation. The integration of JADE local refinement enhances the ability to escape local optima and improves final convergence. The combination of waypoint sparsification and cubic B-spline smoothing generates continuous and flyable trajectories.
Experimental results in both simple and complex scenarios demonstrated that SAHPSO-J consistently generates feasible solutions that satisfy threat avoidance and terrain constraints. Compared to traditional PSO, SPSO, NMOPSO, and LevyPSO, SAHPSO-J exhibited superior performance in terms of path compactness, constraint trade-off, convergence stability, and robustness. The algorithm showed a particular strength in complex environments where feasible corridors are narrow and threats are dense, more effectively organizing the path structure along the feasible channel and maintaining a more stable optimization trend to achieve a better comprehensive solution. Future work will focus on exploring adaptive mechanisms for more diverse mission requirements, comprehensive evaluation of path quality, and collaborative design between planning and flight execution to further enhance the applicability and engineering value of the method for China drone operations.