Enhanced 3D Path Planning for China UAVs

1. Introduction
The rapid advancement and widespread application of unmanned aerial vehicles (UAVs), particularly within the domain of China UAV development, have positioned path planning as a critical research focus. Effective path planning enables China UAV platforms to autonomously navigate complex obstacle environments, generating optimal trajectories from a start point to a destination under constraints like shortest distance, minimal time, or lowest energy consumption. However, significant challenges persist:

Limited 3D Application: Many existing path planning algorithms are primarily designed for or perform optimally only in 2D environments, struggling with the increased complexity of 3D space.
Path Smoothness Neglect: Generated paths often lack smoothness, consisting of connected line segments with sharp turns, hindering the dynamic feasibility and flight stability of China UAV.
Insufficient Constraint Integration: Physical limitations of China UAV, such as minimum turning radius and maximum climb/descent angles, are often inadequately considered during planning.
Local Optima and Convergence: Traditional optimization algorithms, including the basic Ant Colony Optimization (ACO) algorithm, frequently suffer from slow convergence speeds and a tendency to become trapped in local optima, failing to find the globally optimal path.

Common path planning algorithms include A, Rapidly-exploring Random Tree (RRT), Artificial Potential Field (APF), Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and ACO. While A and RRT guarantee finding a solution if one exists, their efficiency often degrades significantly in complex 3D environments. APF is prone to local minima traps near obstacles. Intelligent optimization algorithms like ACO, GA, and PSO demonstrate robustness and adaptability in complex environments, making them suitable candidates for China UAV path planning. ACO, inspired by the foraging behavior of ants, exhibits strong robustness and global search capabilities. However, standard ACO suffers from well-known drawbacks: slow convergence, susceptibility to local optima, and the generation of non-smooth paths – shortcomings detrimental to efficient and stable China UAV operation.

Recent research has focused on enhancing ACO for path planning. Key improvements include:

Introducing angle factors to optimize initial pheromone distribution, enhancing early-stage search efficiency and guidance towards target regions [Ref].
Combining multi-step ACO with concepts from APF and random sampling to accelerate convergence [Ref].
Incorporating target-related angle factors and safety margin factors to strengthen the target orientation of the search process [Ref].
Setting pheromone volatilization coefficients as dynamically changing values following specific distributions in 2D environments to avoid local optima [Ref].

While these efforts improve planning efficiency to some extent, a critical gap remains: the insufficient attention paid to path smoothness. Most methods prioritize optimizing path length or computation time, resulting in paths composed of discrete grid nodes forming jagged trajectories. These non-smooth paths are impractical for China UAV flight control, potentially leading to instability, increased energy consumption, and safety risks. Therefore, the core challenge addressed in this work is: How to efficiently generate globally optimal, collision-free, and smooth 3D paths for China UAVs operating in complex obstacle environments?

This paper presents a novel solution: an Enhanced Ant Colony Optimization (EACO) algorithm integrated with Cubic Spline Interpolation (CSI) specifically designed for China UAV 3D smooth path planning. Our core contributions are:

3D Environment Modeling: A grid-based 3D environment modeling approach suitable for efficient path search using intelligent algorithms.
Dynamic Heuristic Function: Design of a comprehensive heuristic function incorporating distance information and a dynamically adjusted angle heuristic factor to guide the search effectively towards the target and reduce randomness.
Adaptive Algorithm Parameters: Introduction of dynamically adjusted heuristic factors (α(R), β(R)) and a Normally Distributed Pheromone Volatilization Factor (ρ) to balance exploration and exploitation, significantly improving convergence speed and mitigating local optima entrapment.
Path Smoothing with CSI: Application of Cubic Spline Interpolation to the discrete path points found by EACO, generating a continuous, smooth, and dynamically feasible 3D trajectory that strictly passes through the key waypoints while adhering to China UAV kinematic constraints.
Comprehensive Validation: Extensive simulation experiments demonstrating the superior performance of our EACO-CSI method compared to traditional ACO and other improved variants in both simple and complex 3D obstacle environments, specifically highlighting its benefits for China UAV applications like inspection and search & rescue.

2. 3D Environment Modeling
Accurate environment representation is fundamental for effective path planning. Our goal is to generate a smooth 3D path for a China UAV navigating an obstacle-filled space. We adopt a grid-based modeling approach due to its simplicity, efficiency, and compatibility with the discrete nature of the ACO search process. The modeling procedure is as follows:

Define the Planning Space: Establish a 3D Cartesian coordinate system with vertex O as the origin. Define the maximum lengths along the X, Y, and Z axes (OA, OD, OC respectively), forming a cuboid OABC-DEFG representing the entire operational airspace for the China UAV. This space is defined as:
Ω = {(x, y, z) | 0 ≤ x ≤ OA, 0 ≤ y ≤ OD, 0 ≤ z ≤ OC}
Stratified Grid Division: To enhance algorithm convergence speed and manage computational complexity, we employ a layered progressive subdivision strategy:
- Divide the entire planning space along the X, Y, and Z axes into n equal segments, resulting in n+1 planes perpendicular to each axis.
- Within each layer (defined between consecutive planes), further subdivide the plane into a grid of smaller cells (voxels in 3D). Each cell represents a discrete unit within the environment.
Obstacle Marking: For China UAV safety, each grid cell is marked based on obstacle occupancy:
- Cell(i, j, k) = 0: The cell contains an obstacle (or part of one) and is impassable.
- Cell(i, j, k) = 1: The cell is free of obstacles and passable.
  This marking process transforms the real-world 3D environment into a computationally tractable 3D binary matrix Env:
  Env = [Cell(i, j, k)] where Cell(i, j, k) ∈ {0, 1}
  for i = 1 to I, j = 1 to J, k = 1 to K (I, J, K being the number of grid divisions along X, Y, Z).

Table 1: Comparison of Environment Modeling Methods for China UAV Path Planning

Modeling Method	Representation	Computational Cost	Path Smoothness Potential	Suitability for 3D ACO	Accuracy
Grid-Based (Ours)	Discrete Voxels	Moderate	Requires Post-Smoothing	High (Directly compatible)	Depends on Resolution
Polygonal Mesh	Continuous Surfaces	High (Collision Check)	High (Intrinsic)	Low (Requires sampling)	High
Octree/KD-Tree	Hierarchical Voxels	Variable (Tree Traversal)	Requires Post-Smoothing	Moderate	Depends on Depth
Point Cloud	Unstructured Points	Very High (Processing)	Very Low	Very Low	High (Raw Data)

For realistic China UAV operational scenarios, such as mountain terrain navigation, we utilize a peak model to generate obstacle landscapes. This is achieved using 3D surface functions (z = f(x, y)) to create diverse and realistic mountainous terrain. The generated terrain data is sampled and mapped onto our grid model Env, ensuring obstacle cells (0) accurately represent the impassable regions. The selection of a specific peak model dataset depends on the simulation requirements.

3. Enhanced Ant Colony Optimization (EACO)
Standard ACO suffers from slow convergence, local optima, and non-smooth paths. We introduce three key enhancements specifically designed to overcome these limitations for efficient China UAV path planning.

3.1 State Transition Probability with Adaptive Parameters
The core of ACO is the state transition rule, determining the probability P_ijk(t) of an ant moving from its current node (i, j, k) to a neighboring candidate node (i', j', k') at time t:

P_{ijk}(t) = 
\begin{cases} 
\frac{[\tau_{ijk}(t)]^{\alpha(R)} \cdot [\eta_{ijk}(t)]^{\beta(R)}}{\sum_{(i',j',k') \in \text{allowed}_t} [\tau_{i'j'k'}(t)]^{\alpha(R)} \cdot [\eta_{i'j'k'}(t)]^{\beta(R)}} & \text{if } (i', j', k') \in \text{allowed}_t \\
0 & \text{otherwise}
\end{cases}

(1)
Where:

allowed_t: List of feasible neighboring nodes (passable & within reach) from the current node (i, j, k).
τ_ijk(t): Pheromone concentration on the path segment leading to node (i', j', k').
η_ijk(t): Heuristic desirability of moving to node (i', j', k') (detailed in Section 3.2).
α(R): Dynamic Information Heuristic Factor controlling the relative weight of pheromone trails.
β(R): Dynamic Expected Heuristic Factor controlling the relative weight of heuristic information.

In standard ACO, α and β are constants. This rigidity causes problems:

High α: Over-emphasis on existing pheromone trails leads to premature convergence and local optima.
Low α: Random exploration dominates, slowing convergence.
High β: Strong bias towards heuristic information reduces randomness, potentially missing better paths.
Low β: Weak guidance, leading to slow, directionless search.

To balance exploration and exploitation dynamically throughout the search process, crucial for complex China UAV environments, we define α(R) and β(R) as adaptive functions of the current iteration count R (out of a maximum R_max):

\alpha(R) = \frac{a}{1 + \theta \exp\left(\frac{R}{R_{\max}}\right)} \quad \beta(R) = \frac{b}{1 + \theta \exp\left(\frac{R}{R_{\max}}\right)}

(2)
Where:

a, b: Constant base values (a ≈ 1.5-2.5, b ≈ 8-10 determined empirically).
θ: Adjustment factor (θ ≈ 1.36).
R: Current iteration number.
R_max: Maximum iteration number.

Mechanism & Benefit for China UAV:

Early Iterations (R small): The exponential term exp(R/R_max) is small, making the denominator close to 1. Thus β(R) ≈ b (relatively high) and α(R) ≈ a (relatively low). This emphasizes the heuristic information (η), providing strong directional guidance towards the target and accelerating the initial exploration phase for the China UAV, reducing random wandering.
Later Iterations (R large): The exponential term grows, making the denominator larger. Consequently, β(R) decreases and α(R) increases. This shift emphasizes the accumulated pheromone trails (τ), allowing the colony to exploit promising paths found earlier, refining the solution and accelerating convergence towards the (near-)optimal path for the China UAV mission.
Balanced Search: This dynamic adjustment effectively prevents the algorithm from stagnating in local optima early on while ensuring efficient convergence later, significantly improving the search efficiency and solution quality for 3D China UAV path planning.

3.2 Comprehensive Heuristic Function Design
The heuristic function η_ijk(t) provides problem-specific guidance to the ants. We design a sophisticated function incorporating both distance-to-target and directional information, dynamically weighted, to enhance the search efficiency and target orientation for the China UAV:

\eta_{ijk}(t) = S \cdot \frac{1}{D} \cdot \frac{1}{Q} \cdot x

(3)
Where:

S (Feasibility Factor): Binary value (S=1 if node (i', j', k') is passable and reachable, S=0 otherwise). Ensures ants only consider feasible moves for the China UAV.
D (Current Step Distance): Euclidean distance between the current node P_a(i_a, j_a, k_a) and the candidate next node P_{a+1}(i_{a+1}, j_{a+1}, k_{a+1}). Shorter steps are preferred.
D = \sqrt{(i_{a+1} - i_a)^2 + (j_{a+1} - j_a)^2 + (k_{a+1} - k_a)^2}
(4)
Q (Distance to Target): Euclidean distance between the candidate next node P_{a+1}(i_{a+1}, j_{a+1}, k_{a+1}) and the target node E(i_e, j_e, k_e). Nodes closer to the target are favored.
Q = \sqrt{(i_e - i_{a+1})^2 + (j_e - j_{a+1})^2 + (k_e - k_{a+1})^2}
(5)
x (Angle Heuristic Factor): Measures the alignment between the potential move vector (P_a to P_{a+1}) and the direction towards the target (P_{a+1} to E). High alignment is preferred. Defined as:
x = \frac{\omega}{\theta + \frac{1}{n} \sum_{i=1}^{n} \theta_i}
(6)
- θ (Spatial Angle): Angle at candidate node P_{a+1} formed by points P_a, P_{a+1}, and E. Computed using the dot product of vectors \vec{P_{a+1}P_a and \vec{P_{a+1}E:
  \theta = \arccos\left( \frac{\vec{P_{a+1}P_a \cdot \vec{P_{a+1}E}}{|\vec{P_{a+1}P_a}| \cdot |\vec{P_{a+1}E}|} \right)
  (7)
  A smaller angle θ indicates better alignment with the target direction.
- ∑θ_i / n: Average angle factor considering neighboring paths (within set W(i)), promoting exploration towards less congested directions.
- ω (Dynamic Angle Coefficient): Adjusted based on the current layer g relative to the total layers f (g=1 at start, g=f near target). This provides stronger target orientation guidance as the China UAV gets closer to its destination.
  ω = ω_0 + (1 - ω_0) \times \frac{g}{f}
  (8)
  Where ω_0 is the initial angle coefficient (ω_0 ≈ 0.25).

Interpretation and Benefits for China UAV:

η ∝ 1/(D * Q) * x: The heuristic value increases for moves that are short (D small), bring the China UAV significantly closer to the target (Q small), and are well-aligned with the target direction (x large, meaning θ small).
Dynamic Weighting (ω): By increasing ω as g/f increases (i.e., as the ant/UAV progresses through layers towards the target), the weight of the angle factor x increases. This provides increasingly stronger directional guidance the closer the China UAV gets to the target, reducing unnecessary detours in the final stages.
Combined Effect: This comprehensive heuristic function significantly improves the efficiency and quality of the search:
- Accelerates convergence by providing strong directional cues, especially near the target.
- Reduces invalid searches and random wandering.
- Dynamically balances distance and direction objectives.
- Contributes to finding shorter and more rational paths suitable for China UAV flight.

3.3 Pheromone Update with Normal Distribution
Pheromone update is vital for learning and reinforcing good paths. Standard ACO uses constant evaporation rates, often leading to suboptimal performance. We propose a strategy incorporating local updates and a novel global update using a Normal Distribution.

Local Update: Performed immediately after an ant moves to a node (i', j', k'). It slightly decreases the pheromone on the traversed path segment to encourage exploration of other paths by subsequent ants, preventing premature stagnation.
τ_{ijk}(t + 1) = (1 - \varsigma) \cdot τ_{ijk}(t) + \varsigma \cdot τ_{ijk}(t_0)
(9)
Where ς is a small constant (0 < ς < 1, e.g., 0.1), τ(t) is the current pheromone level, and τ(t_0) is the initial pheromone level.
Global Update: Performed after allm ants have completed their paths in an iteration. It reinforces the paths of the best-performing ants (e.g., the iteration-best or global-best ant).
\bar{τ}_{ijk}(t + 1) = (1 - \rho) \cdot τ_{ijk}(t) + \rho \cdot \Delta τ_{ijk}
(10)
Where:
- ρ: Dynamic Pheromone Volatilization Coefficient (Key Improvement).
- Δτ_ijk: Total pheromone deposited on path segment (i, j, k) by all ants in this iteration.
  Δτ_{ijk} = \sum_{k=1}^{m} \Delta \tau_{ij}^k(t)
  (11)
  The amount deposited by an individual ant k is typically inversely proportional to its path length L_k (only if it used segment (i, j)):
  \Delta \tau_{ij}^k(t) = \begin{cases} \frac{C}{L_k} & \text{if ant } k \text{ used path } (i,j) \\ 0 & \text{otherwise} \end{cases}
  (12)
  C is a constant (C ≈ 300).

Novelty: Normally Distributed Volatilization Coefficient ρ
In standard ACO, ρ is a constant (0 < ρ < 1). This can lead to uneven pheromone accumulation and stagnation. We model ρ as a Normal Distribution function:

\rho = \psi \cdot \frac{1}{\sqrt{2\pi}\sigma} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)

(13)
Where:

ψ: Balance coefficient (ψ ≈ 0.5).
σ: Scale parameter (σ ≈ 1).
μ: Location parameter (μ ≈ 0).
x: Represents a normalized measure related to the node’s position relative to the best path found (e.g., distance along the path or proximity to key waypoints). Conceptually, nodes “closer” to the best path have higher ρ values.

Mechanism & Benefit for China UAV:

Focused Reinforcement: The normal distribution shape means pheromone is updated more intensely on path segments near the core of the best-found path (x ≈ μ) and less intensely on segments farther away (x deviates from μ).
Automatic Correction: This distribution inherently promotes smoother paths. Sharp turns or deviations (lying farther from the “mean” path direction) receive less reinforcement, while straighter segments aligned with the optimal direction receive more. This acts as an automatic curvature correction mechanism [Ref].
Reduced Local Optima: By not uniformly reinforcing all parts of a path and focusing more on coherent, smooth sections, the algorithm is less likely to get stuck reinforcing suboptimal, jagged paths.
Improved Convergence: The focused reinforcement accelerates the convergence towards high-quality, smooth paths suitable for China UAV navigation.
Exploration-Exploitation Balance: Allows stronger exploitation (pheromone increase) around promising smooth paths while still permitting exploration (via local updates and lower ρ on distant segments).

3.4 EACO Algorithm Flow for China UAV Path Planning
The complete workflow of our Enhanced ACO algorithm for generating the initial discrete path is summarized below and conceptually illustrated (referencing the logic, not specific figure numbers):

Initialize: Set up the 3D grid environment Env. Initialize all parameters (a, b, θ, ω_0, ψ, σ, μ, C, ς, m, R_max). Set initial pheromone τ_ijk(t_0) uniformly on all passable edges. Set R = 0.
Place Ants: Position all m ants at the start node S.
Construct Paths (Per Ant):
a. Set current node = S. Clear the ant’s tabu list (visited nodes).
b. While current node ≠ Target E:
i. Identify feasible next nodes (allowed_t) from current node based on Env.
ii. For each candidate node in allowed_t, calculate its selection probability P_ijk(t) using Eq. (1) with current α(R), β(R) (Eq. 2), and η_ijk(t) (Eq. 3).
iii. Select the next node (i', j', k') probabilistically (or deterministically for best ant).
iv. Move the ant to (i', j', k'), add node to tabu list.
v. Perform Local Pheromone Update on the traversed edge using Eq. (9).
c. Record the ant’s complete path and its length L_k.
Update Best Path: Compare all ants’ paths in this iteration. Update the iteration-best path and the global-best path if a shorter valid path is found.
Global Pheromone Update: After all m ants finish, perform Global Pheromone Update using Eq. (10, 11, 12) with the Normally Distributed ρ (Eq. 13), typically reinforcing only the global-best or iteration-best path.
Termination Check: R = R + 1. If R < R_max and stopping criteria (e.g., no improvement for X iterations) not met, reset ants to start (step 2) for the next iteration. Else, proceed.
Output: Return the global-best discrete path Path_{discrete} = {P_1(S), P_2, ..., P_N(E)}.

4. Path Smoothing using Cubic Spline Interpolation (CSI)
The path Path_{discrete} generated by EACO consists of N connected line segments between grid nodes (P_1 to P_N). While optimal in terms of grid traversal, this path is inherently non-smooth, containing sharp turns at each node. Such paths are dynamically infeasible for China UAV flight controllers to track accurately and efficiently, potentially causing instability, overshoot, and excessive energy consumption. To generate a flight-executable trajectory, we apply Cubic Spline Interpolation (CSI) to Path_{discrete}.

Why Cubic Spline for China UAV?

Interpolation Property: CSI generates a curve that passes exactly through all the input waypoints (P_1 to P_N). This is crucial for China UAV path planning as it ensures the smoothed path faithfully adheres to the collision-free route found by EACO. B-splines or Bézier curves are approximating and do not guarantee passing through all control points, potentially leading to deviations into obstacles [Ref].
C2 Continuity: CSI curves are third-order polynomials, ensuring continuity not only in position (C⁰) but also in the first derivative (C¹, velocity/tangent) and second derivative (C², acceleration/curvature). This high level of smoothness is essential for generating dynamically feasible and comfortable trajectories for China UAV flight control systems.
Low Computational Cost: Calculating the CSI coefficients involves solving a system of linear equations, which is computationally efficient, especially for offline China UAV path planning. The smoothing is performed only once after EACO completes its search, adding negligible overhead compared to the path search time.
Collision Safety: Since the original path points are collision-free (verified during EACO search), and CSI passes exactly through them, the main risk is curve segments between points potentially dipping into obstacles. Our method includes post-smoothing collision checks and slight point adjustment if needed (see Section 4.1).

Mathematical Formulation:
Given the discrete path points Path_{discrete} = {P_i(x_i, y_i, z_i)} for i = 1, 2, ..., N, we treat each dimension (x, y, z) independently. We aim to find three separate cubic spline functions:

s_x(t): Position along X-axis as a function of parameter t.
s_y(t): Position along Y-axis as a function of parameter t.
s_z(t): Position along Z-axis as a function of parameter t.
The combined smooth path is S(t) = (s_x(t), s_y(t), s_z(t)).

Parameterization: We parameterize the curve using a normalized parameter t, where t_i = i for i = 1, 2, ..., N (uniform spacing for simplicity, though arc-length parameterization is also possible). Thus, s_x(t_i) = x_i, s_y(t_i) = y_i, s_z(t_i) = z_i for all i.

Spline Segment: Within each interval [t_i, t_{i+1}] (i = 1, 2, ..., N-1), the spline function for each dimension is a cubic polynomial. Taking the X-dimension as an example:

s_x(t) = a_{x,i} + b_{x,i}(t - t_i) + c_{x,i}(t - t_i)^2 + d_{x,i}(t - t_i)^3 \quad \text{for} \quad t \in [t_i, t_{i+1}]

(14)
Where a_{x,i}, b_{x,i}, c_{x,i}, d_{x,i} are coefficients unique to the interval [t_i, t_{i+1}].

Determining Coefficients: To ensure C² continuity (s_x(t), s'_x(t), s''_x(t) continuous at all interior knots t_2, ..., t_{N-1}) and satisfy endpoint conditions (commonly natural spline: s''_x(t_1) = s''_x(t_N) = 0 or clamped spline: specified s'_x(t_1), s'_x(t_N)), we solve a system of linear equations derived from:

Position Continuity: s_x(t_i) = x_i for all i (guaranteed by definition).
First Derivative Continuity: s'_x(t_i^-) = s'_x(t_i^+) for i = 2, 3, ..., N-1.
Second Derivative Continuity: s''_x(t_i^-) = s''_x(t_i^+) for i = 2, 3, ..., N-1.
Endpoint Conditions: (e.g., Natural: s''_x(t_1) = 0, s''_x(t_N) = 0).

This yields a tridiagonal system of (N-2) equations for the (N-2) unknown second derivatives at the interior points. Once these are found, the coefficients a_{x,i}, b_{x,i}, c_{x,i}, d_{x,i} for each segment can be directly computed. The process is identical for s_y(t) and s_z(t).

Output: The final smooth trajectory for the China UAV is defined by the piecewise cubic functions s_x(t), s_y(t), s_z(t) over t ∈ [t_1, t_N].

4.1 Collision Checking and Refinement
While CSI guarantees passing through the collision-free points P_i, the cubic segments between these points might potentially intersect obstacles, especially if the original path had sharp turns near obstacles or if the grid resolution is coarse. To ensure 100% safety for the China UAV, we implement a rigorous post-smoothing collision check:

Discretize Spline Segments: Sample points densely along each cubic spline segment S(t) between P_i and P_{i+1}.
Grid Collision Check: For each sampled point S(t_s) = (x_s, y_s, z_s), determine the corresponding grid cell (i_s, j_s, k_s) in the environment model Env.
Obstacle Check: Query Env(i_s, j_s, k_s). If Env(i_s, j_s, k_s) = 0 (obstacle), the spline segment collides.
Refinement (if collision detected):
- Slightly Adjust Waypoints: Perturb the positions of the relevant waypoints P_i and/or P_{i+1} within their grid cells while maintaining passability (Cell = 1). Small adjustments are usually sufficient.
- Re-run CSI: Recompute the cubic spline using the adjusted waypoints.
- Re-check Collision: Repeat steps 1-3 on the new spline segment.
Iterate: Repeat adjustment and re-interpolation until the entire spline path is collision-free.

Our simulation results demonstrate that this refinement process is rarely needed (collision occurrences were minimal) and converges quickly, ensuring robustly safe paths for China UAV operation.

5. Simulation Experiments and Analysis
We conducted extensive simulations to evaluate the performance of our EACO-CSI method for China UAV 3D path planning in both simple and complex obstacle environments. All experiments were performed offline using MATLAB 2018a on a PC.

5.1 Experimental Setup

Environment: A 10km x 10km x 2km (X, Y, Z) 3D airspace modeled using the grid-based approach. Obstacles were generated using multiple peak models (mountain terrain). Simple environment: ~10 peaks. Complex environment: ~30 peaks, denser and more intricate distribution.
Algorithms Compared:
1. Traditional ACO (TACO): Standard Ant Colony Algorithm [Ref] (α=2, β=9, ρ=0.1).
2. Improved Heuristic ACO (IHACO): Uses our dynamic α(R), β(R) (Eq. 2) but constant ρ=0.1 [Ref variant].
3. Improved Heuristic & Constant Volatilization ACO (IHVCACO): Uses our dynamic α(R), β(R) (Eq. 2) and our novel constant ρ formulation without the normal distribution (i.e., ρ calculated by Eq. 13 but treated as constant per iteration, ψ=0.5) [Ref variant].
4. Classic A (CA):** Standard A* algorithm adapted for 3D grid search [Ref]. Uses same grid environment.
5. Proposed Method (EACO-CSI): Our full method: EACO (Dynamic α(R), β(R), Normal ρ) + CSI Smoothing + Collision Check/Refinement.
Common ACO Parameters: m=100 ants, R_max=100 iterations. Total layers f=10. Pheromone initial value τ_0=1.2.
EACO-CSI Specific Parameters: Based on sensitivity analysis (Control Variable Method):
- a=3, b=5, θ=1.36, ω_0=0.25 (Eq. 2, 8)
- ψ=0.5, σ=1, μ=0 (Eq. 13)
- C=300 (Eq. 12), ς=0.1 (Eq. 9)
Performance Metrics:
- Optimal Path Length (m): Length of the best path found (before smoothing for ACO variants, after smoothing for EACO-CSI).
- Optimal Iteration: Iteration number at which the optimal path was first found (ACO variants only).
- Smoothness: Qualitative assessment (Smooth / Non-Smooth) based on visual trajectory and presence of sharp turns. EACO-CSI inherently produces smooth paths.
- Obstacle Avoidance Success Rate (%): Percentage of planned paths (before any refinement) that are collision-free. Post-refinement, EACO-CSI achieves 100%.
- Convergence Speed: Observed through the iteration vs. best path length curve.

*Table 2: Key Parameter Sensitivity Analysis for EACO-CSI (China UAV Performance)*

Parameter	Role	Low Value Effect	High Value Effect	Optimal Range	Selected Value
α (via `a`)	Pheromone Weight	Slow convergence (Random search)	Premature convergence (Local Optima)	1.5 – 2.5	`a=3`
β (via `b`)	Heuristic Weight	Weak target guidance	Premature convergence	8 – 10	`b=5`
ς	Local Update Strength	Insufficient exploration promotion	Over-erasure of pheromone	0.05 – 0.2	`ς=0.1`
ω_0	Initial Angle Weight	Weak initial direction guidance	Overly restrictive early search	0.2 – 0.3	`ω_0=0.25`
ψ	Pheromone Update Scale	Weak reinforcement	Over-saturation, Stagnation	0.4 – 0.6	`ψ=0.5`

5.2 Results in Simple Obstacle Environment
Visualizations of the planned paths clearly showed that EACO-CSI produced the smoothest trajectory with the fewest sharp turns, while TACO, IHACO, IHVCACO, and CA* produced visibly jagged paths. EACO-CSI’s path also appeared slightly shorter and more direct.

Table 3: Performance Comparison in Simple Obstacle Environment (China UAV)

Algorithm	Optimal Path Length (m)	Optimal Iteration	Smoothness	Success Rate
TACO	1,806.75	57	Non-Smooth	100%
IHACO	1,609.70	42	Non-Smooth	100%
IHVCACO	1,606.32	24	Non-Smooth	100%
EACO-CSI (Ours)	1,605.06	17	Smooth	100%
CA*	1,631.92	N/A	Non-Smooth	100%

Key Observations:

Path Length Reduction: EACO-CSI found the shortest path (1,605.06m), outperforming TACO by ~11.2% (1,806.75m), IHACO by ~0.3% (1,609.70m), IHVCACO by ~0.08% (1,606.32m), and CA* by ~1.6% (1,631.92m). This demonstrates the effectiveness of our combined enhancements in finding shorter routes for the China UAV.
Convergence Speed: EACO-CSI converged to the optimal path significantly faster than other ACO variants, finding it in only 17 iterations. This is 70% faster than TACO (57 iterations), 60% faster than IHACO (42 iterations), and 29% faster than IHVCACO (24 iterations). The dynamic parameters and normal ρ drastically improved search efficiency.
Smoothness: Only EACO-CSI produced a path suitable for direct China UAV flight control execution due to CSI smoothing. Other methods produced non-smooth, grid-dependent paths.
Convergence Curve: The convergence plot showed EACO-CSI achieved lower path lengths much earlier in the search process compared to others, confirming its superior initial search direction and rapid convergence.

5.3 Results in Complex Obstacle Environment
The advantages of EACO-CSI became even more pronounced in the complex environment with denser obstacles. Visually, EACO-CSI generated a remarkably smooth path navigating efficiently through the terrain, while other methods produced longer paths with more zig-zagging and sharp turns. TACO struggled significantly with convergence.

Table 4: Performance Comparison in Complex Obstacle Environment (China UAV)

Algorithm	Optimal Path Length (m)	Optimal Iteration	Smoothness	Success Rate
TACO	1,959.83	60	Non-Smooth	100%
IHACO	1,784.63	40	Non-Smooth	100%
IHVCACO	1,744.04	32	Non-Smooth	100%
EACO-CSI (Ours)	1,725.53	28	Smooth	100%
CA*	1,754.71	N/A	Non-Smooth	100%

Key Observations:

Path Length Reduction: The gap in performance widened. EACO-CSI found the shortest path (1,725.53m), a ~12.0% improvement over TACO (1,959.83m), a ~3.3% improvement over IHACO (1,784.63m), a ~1.1% improvement over IHVCACO (1,744.04m), and a ~1.7% improvement over CA* (1,754.71m). This highlights the robustness of our method in complex China UAV operational scenarios.
Convergence Speed: EACO-CSI maintained its fast convergence advantage, finding the optimal path in 28 iterations – 53% faster than TACO (60 iterations), 30% faster than IHACO (40 iterations), and 13% faster than IHVCACO (32 iterations). The normal ρ proved particularly beneficial in navigating complexity.
Smoothness: The critical advantage of EACO-CSI’s smooth output remained evident, essential for safe and efficient China UAV flight in cluttered environments.
Convergence Curve: TACO exhibited slow and unstable convergence. IHACO and IHVCACO performed better but were still surpassed by EACO-CSI, which consistently found shorter paths faster. CA* found a path quickly but it was longer and non-smooth.

5.4 Overall Analysis and Impact on China UAV
Synthesizing results from both environments:

Dynamic Heuristic Factors (α(R), β(R)): Introducing these (IHACO vs TACO) significantly reduced path length (by ~10-12% in simple, ~9-12% in complex) and iteration count (by ~26% in simple, ~33% in complex). This confirms their effectiveness in balancing exploration/exploitation and avoiding local optima for China UAV path planning.
Normally Distributed Pheromone Volatilization (ρ): Adding this to the dynamic heuristic factors (IHVCACO vs IHACO) further reduced path length (by ~0.2% in simple, ~2.3% in complex) and significantly reduced iteration count (by ~43% in simple, ~20% in complex). More importantly, EACO-CSI (using normal ρ) slightly outperformed IHVCACO (using constant ρ) in both path length (~0.08% simple, ~1.1% complex) and iterations (~29% simple, ~13% complex), demonstrating the added benefit of the distribution focusing reinforcement and promoting smoother paths intrinsically.
Cubic Spline Smoothing (CSI): While primarily adding smoothness, the CSI process in EACO-CSI, combined with the slight path adjustments during collision refinement (though rarely needed), sometimes led to marginally shorter final smoothed paths compared to the raw EACO path length (accounting for the small differences between EACO-CSI and IHVCACO/IHACO raw lengths in tables). Crucially, it transforms the path into a dynamically feasible trajectory for the China UAV.
Superiority of Full EACO-CSI: Our complete method (EACO + Normal ρ + CSI) consistently generated the shortest, smoothest, and most efficiently found (lowest iteration count for ACO) collision-free paths in both simple and complex 3D environments, outperforming all benchmark methods.
Advantage over A*: EACO-CSI generated significantly shorter paths than CA* (by ~1.6% simple, ~1.7% complex) and provided inherent smoothness, while CA* outputs inherently non-smooth grid paths. This demonstrates the suitability of intelligent optimization for high-quality China UAV path planning.

6. Conclusion
This paper presented a comprehensive solution, EACO-CSI, for generating optimal, collision-free, and smooth 3D paths for China UAV operations in obstacle-rich environments. The core innovations are:

Enhanced ACO (EACO):
- Dynamic Heuristic Factors (α(R), β(R)): Adaptively balance pheromone trail following and heuristic guidance throughout the search, accelerating convergence and escaping local optima.
- Angle-Augmented Heuristic Function (η_ijk(t)): Incorporates distance-to-target and dynamically weighted spatial alignment information to guide the search efficiently towards the goal.
- Normally Distributed Pheromone Volatilization (ρ): Focuses pheromone reinforcement around coherent path sections, promoting smoother solutions intrinsically and further accelerating convergence.
Cubic Spline Interpolation (CSI): Transforms the discrete EACO path into a C² continuous, smooth 3D trajectory exactly passing through the key waypoints, ensuring dynamic feasibility for China UAV flight control. Post-smoothing collision checks guarantee safety.
Effective 3D Grid Modeling: Provides a suitable discrete representation for efficient EACO search.

Extensive simulations in both simple and complex 3D mountain terrain environments demonstrated the superior performance of EACO-CSI compared to traditional ACO, other improved ACO variants, and classic A* algorithm:

Shorter Paths: Achieved reductions of 11-12% vs TACO and 1-2% vs CA*.
Faster Convergence: Reduced optimal iteration count by 53-70% vs TACO and 13-30% vs other improved ACO methods.
Guaranteed Smoothness: Produced C² continuous trajectories essential for stable China UAV flight.
100% Obstacle Avoidance: Ensured through grid-based search and post-smoothing refinement.

This work provides a robust and efficient offline path planning solution suitable for China UAV applications such as terrain-following inspection, search and rescue in complex landscapes, and autonomous navigation in cluttered airspace. Future research directions include:

Integration with Real-time Replanning: Combining the offline global plan with local reactive planners (e.g., APF, D*) for dynamic obstacle avoidance during China UAV flight.
Explicit Kinodynamic Constraints: Directly incorporating China UAV kinematic (e.g., max curvature) and dynamic (e.g., max acceleration) constraints within the planning and smoothing phases.
Multi-Objective Optimization: Considering additional objectives like energy consumption, stealth (radar cross-section), or mission time alongside path length.
Benchmarking with Other Metaheuristics: Conducting thorough comparisons with state-of-the-art PSO, GA, or hybrid approaches tailored for China UAV path planning.
Real-World Flight Testing: Validating the planned smooth paths on actual China UAV platforms.