In the rapidly evolving field of automated warehousing and logistics, the path planning of UAV drones has become a critical optimization problem. UAV drones operating in complex environments must generate short, safe, and smooth trajectories while avoiding obstacles and satisfying multiple constraints such as flight altitude, turning angles, and energy consumption. Many existing studies have tackled this problem using metaheuristic algorithms, yet they often suffer from premature convergence, poor global search capability, or high computational cost. To address these limitations, we propose an improved whale optimization algorithm (IWOA) that integrates Tent chaotic mapping initialization, nonlinear dynamic parameter adjustment, and an adaptive inertia weight factor. Our approach is specifically designed to enhance the path planning performance of UAV drones in two-dimensional, three-dimensional, and warehouse-like environments.
In this paper, we first outline the standard whale optimization algorithm (WOA) and its inherent weaknesses. Then we detail the three key improvements of IWOA: Tent chaotic initialization to preserve population diversity, nonlinear adjustment of the coefficient \(C\) and the convergence factor \(\alpha\), and a cosine-based adaptive weight \(\gamma\). These modifications allow IWOA to balance exploration and exploitation effectively, leading to superior convergence speed and solution quality. We evaluate the algorithm on standard benchmark functions and compare it with six other common optimizers. Subsequently, we build realistic simulation models for UAV drones path planning, including terrain, obstacles, and kinematic constraints. The IWOA-based planner computes the optimal path that minimizes a weighted sum of threat, path length, and altitude costs. Extensive simulations demonstrate that IWOA significantly outperforms WOA, particle swarm optimization (PSO), genetic algorithm (GA), artificial bee colony (ABC), ant colony optimization (ACO), A* algorithm, and other improvements in terms of path length, computation time, and solution robustness. The results confirm that IWOA is a highly effective tool for autonomous navigation of UAV drones in challenging environments.
Standard Whale Optimization Algorithm
The WOA mimics the hunting behavior of humpback whales, which includes three phases: searching for prey, encircling prey, and bubble-net attacking. The mathematical model is as follows.
During the search phase, a whale updates its position based on a randomly selected member of the population:
$$
D_1 = |C \cdot X_{\text{rand}}(t) – X(t)|
$$
$$
X(t+1) = X_{\text{rand}}(t) – A \cdot D_1
$$
where \(t\) is the current iteration, \(X(t)\) is the current position vector, \(X_{\text{rand}}(t)\) is a random position, and \(A\) and \(C\) are coefficient vectors defined as:
$$
A = 2\alpha r – \alpha, \quad C = 2r, \quad \alpha = 2\left(1 – \frac{t}{T}\right)
$$
with \(r \in [0,1]\) and \(T\) the maximum number of iterations.
In the encircling phase, the whale moves toward the current best solution \(X^*(t)\):
$$
D_2 = |C \cdot X^*(t) – X(t)|
$$
$$
X(t+1) = X^*(t) – A \cdot D_2
$$
For bubble-net attacking, a spiral equation is used:
$$
D_3 = |X^*(t) – X(t)|
$$
$$
X(t+1) = D_3 \cdot e^{bl} \cdot \cos(2\pi l) + X^*(t)
$$
where \(b\) is a constant and \(l \in [-1,1]\). The complete WOA position update rule with probability \(\rho \in [0,1]\) is:
$$
X(t+1) =
\begin{cases}
X^*(t) – A D_2, & (\rho < 0.5) \land (|A| < 1) \\
D_3 e^{bl} \cos(2\pi l) + X^*(t), & \rho \ge 0.5 \\
X_{\text{rand}}(t) – A D_1, & (\rho < 0.5) \land (|A| \ge 1)
\end{cases}
$$
Although WOA is simple and has few parameters, its random initialization and linear parameter settings often lead to premature convergence and poor solution quality in complex multimodal optimization problems such as UAV drones path planning.
Improved Whale Optimization Algorithm (IWOA)
To overcome the limitations of standard WOA, we propose three major improvements: Tent chaotic map initialization, nonlinear dynamic adjustment of parameters, and an adaptive inertia weight.
Tent Chaotic Map Initialization
Instead of random initialization, we employ the Tent chaotic map to generate the initial population. This map is defined as:
$$
x_{i+1} =
\begin{cases}
2x_i, & x_i < 0.5 \\
2(1 – x_i), & x_i \ge 0.5
\end{cases}
$$
The chaotic sequence provides better coverage of the search space, enhances population diversity, and helps the algorithm escape local optima from the very beginning. The following diagram illustrates the Tent mapping initialization.
Nonlinear Dynamic Adjustment of C
In the standard WOA, \(C = 2r\) is a random number that does not adapt to the search stage. We propose adjusting \(C\) nonlinearly with iterations:
$$
C = 2 – \left(\frac{t}{T}\right)^3
$$
This modification ensures that \(C\) stays near 2 in early iterations (promoting global exploration), decreases rapidly in mid-stages to facilitate transition, and approaches 1 in later stages for fine local exploitation.
Nonlinear Adjustment of Convergence Factor α
The convergence factor \(\alpha\) controls the magnitude of \(A\). To improve the ability to jump out of local minima, we replace the linear decay with a cosine decreasing function:
$$
\alpha =
\begin{cases}
2\cos\left(\frac{\pi}{2} \cdot \frac{t}{T}\right), & P_\alpha(t) \le P_\alpha \\
2n^2\sin(\pi n), & P_\alpha(t) > P_\alpha
\end{cases}
$$
where \(P_\alpha\) is a predefined probability and \(n = \text{rand}(0,1)\). This nonlinear schedule allows a larger \(\alpha\) in the early phase for fast convergence and a smaller \(\alpha\) later for high precision.
Adaptive Inertia Weight
We introduce a cosine-based inertia weight \(\gamma\) to balance global and local search:
$$
\gamma = \cos\left(\frac{\pi}{2} \cdot \frac{t}{T} + 1\right) + 1
$$
The weight is high in early iterations (large search scope), moderate in middle, and low in later iterations (fine-tuning). The updated position formula becomes:
$$
X(t+1) =
\begin{cases}
X^*(t) – \gamma A D_2, & (\rho < 0.5) \land (|A| < 1) \\
\gamma D_3 e^{bl} \cos(2\pi l) + X^*(t), & \rho \ge 0.5 \\
X_{\text{rand}}(t) – \gamma A D_1, & (\rho < 0.5) \land (|A| \ge 1)
\end{cases}
$$
We compared three weight strategies (fixed, linear, adaptive) on three categories of benchmark functions. The results are summarized in Table 1.
| Function Type | Weight Strategy | Best Solution | Std Dev | Convergence Iterations | EDBI |
|---|---|---|---|---|---|
| Unimodal | Fixed | 3.21E-5 | 2.1E-6 | 182 | 0.45 |
| Linear | 2.87E-5 | 1.8E-6 | 165 | 0.38 | |
| Adaptive (IWOA) | 1.02E-6 | 5.3E-8 | 121 | 0.22 | |
| Multimodal | Fixed | 4.56E-3 | 3.2E-4 | 327 | 0.87 |
| Linear | 3.12E-3 | 2.7E-4 | 267 | 0.66 | |
| Adaptive (IWOA) | 8.74E-4 | 9.1E-5 | 231 | 0.41 | |
| Composite | Fixed | 1.24E-2 | 1.1E-3 | 496 | 1.12 |
| Linear | 9.87E-3 | 8.4E-4 | 421 | 0.95 | |
| Adaptive (IWOA) | 5.43E-3 | 4.7E-4 | 378 | 0.65 |
As shown, the adaptive inertia weight (used in IWOA) achieves the smallest best solution, lowest standard deviation, and fastest convergence across all function types. The exploration‑development balance index (EDBI) is also significantly reduced, indicating a better trade-off between global and local search.
Path Planning Model for UAV Drones
We formulate the path planning problem for UAV drones as a constrained optimization task. The goal is to find a sequence of waypoints from a start position to a target position while avoiding obstacles and respecting physical limitations.
Environment Modeling
The terrain elevation is given by:
$$
z(x,y) = \sin(y + a) + b’ \sin x + c \cos(d \sqrt{x^2 + y^2}) + e \cos y + f \sin(f \sqrt{x^2 + y^2}) + g \cos y
$$
Obstacles (e.g., buildings, shelves) are modeled as Gaussian peaks:
$$
z(x,y) = \sum_{i’} h_{i’} \exp\left( -\left(\frac{x – y_{i’}}{x_{s i’}}\right)^2 – \left(\frac{y – x_{i’}}{y_{s i’}}\right)^2 \right)
$$
where \(h_{i’}\) is the obstacle height, \((x_{i’}, y_{i’})\) is its center, and \(x_{s i’}, y_{s i’}\) control the slope.
Trajectory Smoothing
To generate a flyable path, we use cubic B-spline interpolation. The B-spline curve is defined as:
$$
P(u) = \sum_{i’=0}^{n} d_{i’} N_{i’,k}(u)
$$
where \(d_{i’}\) are control points and \(N_{i’,k}(u)\) are basis functions computed recursively:
$$
N_{i’,0}(u) =
\begin{cases}
1, & u_i \le u \le u_{i+1} \\
0, & \text{otherwise}
\end{cases}
$$
$$
N_{i’,k}(u) = \frac{u – u_i}{u_{i+k} – u_i} N_{i’,k-1}(u) + \frac{u_{i+k+1} – u}{u_{i+k+1} – u_{i+1}} N_{i’+1,k-1}(u)
$$
Constraints
We impose three constraints on the path: flight altitude, turning angle, and pitch angle.
Altitude constraint: The height \(z_i\) of each waypoint must lie within safe bounds:
$$
H_{\min} \le z_i \le H_{\max}
$$
Turning angle constraint: The horizontal turning angle \(\beta_i\) must not exceed the maximum allowable value \(\beta_{\max}\):
$$
\beta_i = \arccos\left( \frac{(x_i – x_{i-1})(x_{i+1} – x_i) + (y_i – y_{i-1})(y_{i+1} – y_i)}{\sqrt{(x_i – x_{i-1})^2 + (y_i – y_{i-1})^2} \cdot \sqrt{(x_{i+1} – x_i)^2 + (y_{i+1} – y_i)^2}} \right) \le \beta_{\max}
$$
Pitch angle constraint: The vertical pitch angle \(\mu_i\) must be within limits:
$$
\mu_i = \arccos\left( \frac{z_i – z_{i-1}}{\sqrt{(x_i – x_{i-1})^2 + (y_i – y_{i-1})^2}} \right) \le \mu_{\max}
$$
Cost Functions
The overall path cost is a weighted sum of three components: threat cost \(F_1\), path length cost \(F_2\), and altitude change cost \(F_3\).
Threat cost: Penalizes proximity to obstacles.
$$
F_1 = \sum_{t=1}^{T} q_t
$$
$$
q_t =
\begin{cases}
\frac{d_{t,t-1}}{\cos(\pi / (3d_{\min})) \cdot d_{t,b}}, & 0 \le d_{t,b} \le d_{\min} \\
\frac{\sqrt{3}}{3} \frac{d_{t,t-1}}{\cos(\pi / (6d_{\min})) \cdot d_{t,b}}, & d_{\min} \le d_{t,b} \le \sqrt{3}d_{\min} \\
0, & d_{t,b} > \sqrt{3}d_{\min}
\end{cases}
$$
where \(d_{t,b}\) is the distance from the UAV drone to the nearest obstacle at waypoint \(t\), and \(d_{t,t-1}\) is the segment length between waypoints \(t\) and \(t-1\).
Path length cost: Minimizes total distance:
$$
F_2 = \sum_{t=1}^{T} \sqrt{(x_t – x_{t-1})^2 + (y_t – y_{t-1})^2 + (z_t – z_{t-1})^2}
$$
Altitude change cost: Penalizes unnecessary altitude variations to save energy and maintain stability:
$$
F_3 = \sum_{t=1}^{T} h_t
$$
$$
h_t = \frac{|x_{t,t-1}|}{2} \left(1 + \cos\left(2\pi \frac{z_t – H_{\min}}{H_{\max} – H_{\min}}\right)\right), \quad x_{t,t-1} = z_t – z_{t-1}
$$
The overall objective function to minimize is:
$$
\min F = \alpha_1 F_1 + \alpha_2 F_2 + \alpha_3 F_3
$$
Using the analytic hierarchy process (AHP), we set the weights as \(\alpha_1 = 0.3531\), \(\alpha_2 = 0.4519\), \(\alpha_3 = 0.1950\).
Simulation and Validation
We conducted extensive simulations to evaluate the performance of IWOA for UAV drones path planning. The simulations were performed in MATLAB, where we built 2D, 3D, and warehouse environment models. We compared IWOA with WOA, improved artificial bee colony (IABC), improved genetic algorithm (IGA), and other algorithms. The results are summarized in the following tables.
2D Environment
In a 2D random obstacle field, IWOA produced smoother and shorter paths than all competitors. Table 2 shows the quantitative comparison.
| Algorithm | Path Length (m) | Computation Time (s) |
|---|---|---|
| WOA | 12.45 | 3.21 |
| IABC | 11.53 | 2.77 |
| IGA | 11.73 | 2.84 |
| IWOA | 11.24 | 2.44 |
IWOA reduced path length by 9.74% compared to WOA, by 2.59% compared to IABC, and by 4.2% compared to IGA. Computation time was reduced by 23.96%, 11.82%, and 14.04%, respectively. On average, IWOA achieved a 5.51% reduction in distance and 16.6% reduction in time.
3D Environment
In a 3D environment with spherical obstacles, IWOA successfully avoided all collisions while WOA failed in some cases. Table 3 presents the results.
| Algorithm | Path Length (m) | Computation Time (s) |
|---|---|---|
| WOA | 28.31 | 5.67 |
| IABC | 26.38 | 5.43 |
| IGA | 26.68 | 5.48 |
| IWOA | 25.46 | 5.28 |
IWOA path length was 10.07% shorter than WOA, 3.49% shorter than IABC, and 4.61% shorter than IGA. Time savings were 6.88%, 2.72%, and 3.86%, respectively. Average improvements across the board: 6.06% distance reduction and 4.49% time reduction.
Warehouse Environment
To mimic a realistic storage facility, we placed obstacles in a regular grid pattern. IWOA again produced the best paths, while WOA collided and IGA generated sharp turns unsuitable for real UAV drones. Table 4 summarizes the results.
| Algorithm | Path Length (m) | Computation Time (s) |
|---|---|---|
| IABC | 18.77 | 4.22 |
| IWOA | 17.44 | 3.87 |
IWOA shortened the path by 7.09% and reduced computation time by 8.29% compared to the best alternative (IABC).
Conclusion
We have developed an improved whale optimization algorithm (IWOA) tailored for the path planning of UAV drones. By introducing Tent chaotic initialization, nonlinear adjustments of parameters \(C\) and \(\alpha\), and a cosine adaptive inertia weight, IWOA significantly enhances global search capability, convergence speed, and solution quality. Extensive simulations in 2D, 3D, and warehouse environments demonstrate that IWOA consistently outperforms WOA, IABC, IGA, and other popular algorithms. The planned paths are shorter, smoother, and computed faster, making IWOA a practical and efficient solution for autonomous navigation of UAV drones in complex real-world scenarios. Future work will focus on real-time implementation and testing on actual UAV drones hardware using Simulink and embedded controllers to further validate robustness in dynamic environments.