The efficient operation of Unmanned Aerial Vehicles (UAVs) across diverse fields—military reconnaissance, logistics, environmental monitoring, agriculture, and emergency response—critically hinges on effective path planning. This ensures mission safety, timeliness, collision avoidance, minimized energy consumption, and adaptability within complex three-dimensional (3D) environments. Traditional methods, such as Artificial Potential Fields and A* algorithms, often struggle with computational complexity, susceptibility to local optima, and limited dynamic adaptability in 3D spaces. Consequently, metaheuristic optimization algorithms have gained prominence due to their global search capabilities and adaptability. This work presents a novel Improved Mountaineering Team-Based Optimization (IMTBO) algorithm specifically designed to address the challenges of 3D path planning for Unmanned Aerial Vehicles, overcoming limitations like slow convergence and local optima entrapment observed in the standard MTBO algorithm.
1. Mathematical Model for Unmanned Aerial Vehicle 3D Path Planning
The core objective is to find an optimal or feasible flight path connecting a start point S=(xs,ys,zs)S=(xs,ys,zs) to a goal point G=(xg,yg,zg)G=(xg,yg,zg) within a defined 3D operational airspace for the Unmanned Aerial Vehicle. The solution must satisfy constraints related to obstacle avoidance, flight dynamics, and mission requirements.
1.1 Environment Modeling
The operational environment is modeled using a Digital Elevation Model (DEM), integrating terrain elevation data and obstacle information. The airspace boundaries are defined as:
- x∈[xmin,xmax]x∈[xmin,xmax]
- y∈[ymin,ymax]y∈[ymin,ymax]
- z∈[zmin,zmax]z∈[zmin,zmax]
Obstacles are modeled as spheres or cylinders, each characterized by its center coordinates and radius rkrk, for k=1,2,…,Mk=1,2,…,M, where MM is the total number of obstacles.
1.2 Path Representation (Encoding)
A flight path is represented as a sequence of NN navigation points: P1,P2,…,PNP1,P2,…,PN, where Pi=(xi,yi,zi)Pi=(xi,yi,zi). This sequence is encoded into a solution vector XX for optimization algorithms:X=[x1,y1,z1,x2,y2,z2,…,xN,yN,zN]X=[x1,y1,z1,x2,y2,z2,…,xN,yN,zN]
Each coordinate must lie within the defined airspace boundaries.
1.3 Objective Function
The quality of a path is evaluated using a weighted sum objective function JJ that combines three critical metrics for Unmanned Aerial Vehicle operations:
- Path Length (JpathJpath): The total Euclidean distance traveled.Jpath=∑i=1N−1(xi+1−xi)2+(yi+1−yi)2+(zi+1−zi)2Jpath=i=1∑N−1(xi+1−xi)2+(yi+1−yi)2+(zi+1−zi)2
- Height Cost (JheightJheight): Measures the root mean square (RMS) of height changes relative to a nominal height or terrain, penalizing excessive altitude variations for smoother flight and energy efficiency. Let zrefzref be a reference height (e.g., terrain height at (xi,yi)(xi,yi) or a constant desired cruise height).Jheight=1N∑i=1N(zi−zref)2Jheight=N1i=1∑N(zi−zref)2
- Path Smoothness (JsmoothJsmooth): Measures the cumulative turning angles between consecutive path segments, promoting smoother trajectories essential for Unmanned Aerial Vehicle maneuverability.Jsmooth=∑i=2N−1θiJsmooth=i=2∑N−1θiwhere θiθi is the angle between vectors Pi−1Pi→Pi−1Pi and PiPi+1→PiPi+1.
The composite objective function is:J=w1⋅Jpath+w2⋅Jheight+w3⋅JsmoothJ=w1⋅Jpath+w2⋅Jheight+w3⋅Jsmooth
where w1w1, w2w2, and w3w3 are weighting coefficients (w1+w2+w3=1w1+w2+w3=1) reflecting the relative importance of each criterion for the specific Unmanned Aerial Vehicle mission.
1.4 Constraints
The path must adhere to several constraints:
- Obstacle Avoidance: Every path point PiPi must maintain a minimum safe distance rsafersafe from all obstacles.mink=1M∥Pi−Pobsk∥≥rsafe+rk,∀i=1,…,Nk=1minM∥Pi−Pobsk∥≥rsafe+rk,∀i=1,…,NA penalty function is incorporated into the objective function to handle violations:Penalty=p⋅∑i=1N∑k=1Mmax(1−∥Pi−Pobsk∥rsafe+rk,0)Penalty=p⋅i=1∑Nk=1∑Mmax(1−rsafe+rk∥Pi−Pobsk∥,0)L=J+PenaltyL=J+Penaltywhere pp is a large penalty coefficient. LL replaces JJ as the fitness function during optimization.
- Flight Altitude Constraint:zmin≤zi≤zmax,∀i=1,…,Nzmin≤zi≤zmax,∀i=1,…,N
- Flight Area Constraint:xmin≤xi≤xmax,ymin≤yi≤ymax,∀i=1,…,Nxmin≤xi≤xmax,ymin≤yi≤ymax,∀i=1,…,N
- Dynamic Constraints (Simplified): Represent limits on the Unmanned Aerial Vehicle’s velocity and acceleration between points.vmin≤∥Pi+1−Pi∥Δt≤vmax,amin≤∥(Pi+1−Pi)/Δt−(Pi−Pi−1)/Δt∥Δt≤amaxvmin≤Δt∥Pi+1−Pi∥≤vmax,amin≤Δt∥(Pi+1−Pi)/Δt−(Pi−Pi−1)/Δt∥≤amaxThese are often handled implicitly through smoothness constraints or explicitly via penalty terms.
Table 1: Summary of Constraints for Unmanned Aerial Vehicle Path Planning
| Constraint Type | Mathematical Formulation | Handling Mechanism |
|---|---|---|
| Obstacle Avoidance | mink∥Pi−Pobsk∥≥rsafe+rkmink∥Pi−Pobsk∥≥rsafe+rk | Penalty Function added to JJ: L=J+p⋅∑i∑kmax(1−dikrsafe+rk,0)L=J+p⋅∑i∑kmax(1−rsafe+rkdik,0) |
| Flight Altitude | zmin≤zi≤zmaxzmin≤zi≤zmax | Direct boundary constraint during solution generation/repair. Penalty function. |
| Flight Area | xmin≤xi≤xmax,ymin≤yi≤ymaxxmin≤xi≤xmax,ymin≤yi≤ymax | Direct boundary constraint during solution generation/repair. Penalty function. |
| Velocity (Simplified) | vmin≤∥Pi+1−Pi∥/Δt≤vmaxvmin≤∥Pi+1−Pi∥/Δt≤vmax | Implicit via smoothness, explicit via penalty on segment length deviation. |
| Acceleration (Simplified) | amin≤∥Δvi∥/Δt≤amaxamin≤∥Δvi∥/Δt≤amax | Implicit via smoothness (JsmoothJsmooth), explicit via penalty on angle change. |
2. Standard Mountaineering Team-Based Optimization (MTBO)
The MTBO algorithm is a population-based metaheuristic inspired by the collaborative ascent strategies of a mountaineering team aiming to reach a summit (the global optimum). The population consists of NN members (mountaineers), including one leader (the best-performing member).
2.1 Initialization
The initial positions Xi=[xi1,xi2,…,xiD]Xi=[xi1,xi2,…,xiD] of the NN team members within the DD-dimensional search space (where D=3ND=3N for our path encoding) are generated randomly:xij=lj+rand(0,1)⋅(uj−lj),i=1,…,N;j=1,…,Dxij=lj+rand(0,1)⋅(uj−lj),i=1,…,N;j=1,…,D
where ljlj and ujuj are the lower and upper bounds for dimension jj, and rand(0,1)rand(0,1) is a uniform random number.
2.2 Phases and Position Update
MTBO employs four distinct phases, selected probabilistically during each iteration for each member:
- Collaborative Climbing (CC): Members learn from the leader (XleaderXleader) and a randomly selected better-performing member (XrXr).Xinew=Xi+r2⋅(Xleader−Xi)+r3⋅(Xr−Xi)Xinew=Xi+r2⋅(Xleader−Xi)+r3⋅(Xr−Xi)where r2,r3∼rand(0,1)r2,r3∼rand(0,1).
- Disaster Threat (DT): Simulates avoiding hazardous areas (local optima), represented by the worst member’s position (XworstXworst).Xinew=Xi−r4⋅(Xworst−Xi)Xinew=Xi−r4⋅(Xworst−Xi)where r4∼rand(0,1)r4∼rand(0,1).
- Coordinated Defense (CD): Simulates team cohesion by moving towards the team’s centroid (Xˉ=1N∑k=1NXkXˉ=N1∑k=1NXk).Xinew=Xi−r5⋅(Xˉ−Xi)Xinew=Xi−r5⋅(Xˉ−Xi)where r5∼rand(0,1)r5∼rand(0,1).
- Member Update (MU): Simulates losing members (e.g., to avalanches) and recruiting new ones. The worst member is replaced by a new member generated randomly within the search space.Xworstnew=[xrand1,xrand2,…,xrandD],xrandj=lj+rand(0,1)⋅(uj−lj)Xworstnew=[xrand1,xrand2,…,xrandD],xrandj=lj+rand(0,1)⋅(uj−lj)
After each position update, the fitness LL (penalized objective) is evaluated. The leader (XleaderXleader) and worst member (XworstXworst) are identified based on these fitness evaluations. The process repeats until a termination criterion (e.g., maximum iterations TT) is met.
3. Improved MTBO (IMTBO) for Unmanned Aerial Vehicle Path Planning
The standard MTBO algorithm exhibits limitations in Unmanned Aerial Vehicle path planning, particularly concerning initial population diversity, balance between exploration and exploitation, and escaping local optima. IMTBO incorporates four key strategies to enhance performance.
3.1 Tent Chaotic Mapping for Initialization
Replaces random initialization to ensure better coverage of the search space. The Tent map generates chaotic sequences with high ergodicity:xij(0)={xi,j−1(0)/αif xi,j−1(0)∈[0,α)(1−xi,j−1(0))/(1−α)if xi,j−1(0)∈[α,1]xij(0)={xi,j−1(0)/α(1−xi,j−1(0))/(1−α)if xi,j−1(0)∈[0,α)if xi,j−1(0)∈[α,1]
where α∈(0,1)α∈(0,1) (typically α=0.5α=0.5), j=1,…,Dj=1,…,D, and xi0(0)xi0(0) is a random seed in [0,1][0,1]. The chaotic values xij(0)xij(0) are scaled to the search space:xij=lj+xij(0)⋅(uj−lj)xij=lj+xij(0)⋅(uj−lj)
This generates half of the initial population (N/2N/2 members).
3.2 Refraction Reverse Learning (RRL) for Enhanced Diversity
Generates the other half of the initial population based on the Tent-mapped individuals using principles of light refraction. For a Tent-mapped member XiXi, its RRL counterpart XiRRLXiRRL is:xijRRL=lj+uj2+lj+uj2k−xijkxijRRL=2lj+uj+2klj+uj−kxij
where k>0k>0 is a refraction coefficient (typically k=0.75k=0.75). The full initial population is the union of the Tent-mapped set and the RRL set. This strategy significantly boosts initial exploration capability.
3.3 Sine Cosine Strategy (SCS) for Disaster Threat Phase
Replaces the standard DT update (Eq. 12) to dynamically balance exploration and exploitation. The SCS leverages the oscillatory nature of sine and cosine functions:Xinew={w⋅Xi+r6⋅sin(r7)⋅∣r8⋅Xleader−Xi∣if r9<Sw⋅Xi+r6⋅cos(r7)⋅∣r8⋅Xleader−Xi∣if r9≥SXinew={w⋅Xi+r6⋅sin(r7)⋅∣r8⋅Xleader−Xi∣w⋅Xi+r6⋅cos(r7)⋅∣r8⋅Xleader−Xi∣if r9<Sif r9≥S
The components are defined as follows:
- ww: Nonlinear decreasing weight factor, controlling reliance on current position. Increases exploitation over time.w=et−1e−1w=e−1et−1where t=current iteration/Tt=current iteration/T, T=max iterationsT=max iterations.
- r6r6: Adaptive step size factor, decreases nonlinearly to transition from exploration to exploitation.r6=[1−(tT)]1/ar6=[1−(Tt)]1/awhere a>1a>1 is a tuning parameter (e.g., a=1.6a=1.6).
- r7∈[0,2π]r7∈[0,2π], r8∈[0,2π]r8∈[0,2π], r9∈[0,1]r9∈[0,1]: Random numbers.
- S∈[0,1]S∈[0,1]: Threshold parameter (e.g., S=0.7S=0.7) controlling the switch probability between sine and cosine terms.
The SCS dynamically adjusts the search direction and step size around the leader’s position, enhancing the ability to escape local optima inherent in complex Unmanned Aerial Vehicle path planning scenarios.
3.4 Gaussian Mutation for Member Update Phase
Replaces the random re-initialization in the standard MU phase (Eq. 13). Instead of generating a completely new random member, the worst member undergoes a guided perturbation using Gaussian mutation:Xworstnew=Xworst⊙(1+K)Xworstnew=Xworst⊙(1+K)
where ⊙⊙ denotes element-wise multiplication, and K=[k1,k2,…,kD]K=[k1,k2,…,kD] is a vector of random numbers drawn from a Gaussian (Normal) distribution with mean μ=0μ=0 and a small standard deviation σσ (e.g., σ=0.1⋅(uj−lj)σ=0.1⋅(uj−lj)):kj∼N(0,σ2)kj∼N(0,σ2)
This strategy applies small, localized perturbations to the worst solution, facilitating fine-tuning and exploitation around potentially promising regions, which is crucial for refining Unmanned Aerial Vehicle paths late in the optimization process.
3.5 IMTBO Workflow for Unmanned Aerial Vehicle Path Planning
The complete IMTBO procedure for solving the Unmanned Aerial Vehicle 3D path planning problem is outlined below:
- Environment Setup: Define airspace boundaries [xmin,xmax][xmin,xmax], [ymin,ymax][ymin,ymax], [zmin,zmax][zmin,zmax], start SS, goal GG, obstacle parameters (Pobsk,rk,rsafePobsk,rk,rsafe), navigation points NN, weights w1,w2,w3w1,w2,w3, penalty coefficient pp.
- IMTBO Initialization: Set algorithm parameters: population size NpopNpop, max iterations TT, αα, kk, SS, aa, σσ. Generate initial population using Tent Chaotic Mapping and Refraction Reverse Learning.
- Fitness Evaluation: Calculate the penalized objective LL (Eq. 3) for each member. Identify XleaderXleader and XworstXworst.
- Main Loop (Iteration t=1t=1 to TT):
- For each member ii:
- Select an update phase probabilistically (CC, DT-SCS, CD, MU-Gaussian).
- Generate new position XinewXinew using the selected phase’s update rule (Eq. 11, 15, 13, or 18).
- Apply boundary constraints if XinewXinew exceeds bounds.
- Evaluate fitness LnewLnew of XinewXinew.
- Greedy Selection: If Lnew<LiLnew<Li (minimization), update Xi=XinewXi=Xinew and Li=LnewLi=Lnew.
- Update XleaderXleader (best member) and XworstXworst (worst member).
- For each member ii:
- Termination: Output XleaderXleader (representing the best-found path P1,…,PNP1,…,PN) and its fitness LleaderLleader.
Table 2: Key IMTBO Parameters and Typical Values for Unmanned Aerial Vehicle Path Planning
| Parameter | Symbol | Description | Typical Value/Range |
|---|---|---|---|
| Population Size | NpopNpop | Number of mountaineers (solutions) in the team. | 20-100 |
| Maximum Iterations | TT | Stopping criterion. | 100-500 |
| Path Points | NN | Number of navigation points defining the path. | 5-20 |
| Weight (Length) | w1w1 | Importance of minimizing total path distance. | 0.5-0.8 |
| Weight (Height) | w2w2 | Importance of minimizing altitude variation. | 0.1-0.3 |
| Weight (Smoothness) | w3w3 | Importance of path smoothness (low turning angles). | 0.1-0.3 |
| Penalty Coefficient | pp | Large constant penalizing obstacle constraint violations. | 103−106103−106 |
| Safe Distance | rsafersafe | Minimum clearance required between UAV and obstacles. | Mission-dependent |
| Tent Parameter | αα | Chaotic map coefficient controlling bifurcation. | 0.5 |
| Refraction Coefficient | kk | Controls the “bending” magnitude in RRL. k>1k>1 bends towards center. | 0.5-1.0 (e.g., 0.75) |
| SCS Threshold | SS | Probability threshold for switching between sine/cosine update in DT phase. | 0.5-0.8 (e.g., 0.7) |
| SCS Decay Parameter | aa | Controls the nonlinear decay rate of step size factor r6r6. Larger aa = slower decay. | >1 (e.g., 1.6) |
| Gaussian Std. Dev. | σσ | Controls the magnitude of mutation applied in the MU phase. | 0.1⋅(uj−lj)0.1⋅(uj−lj) |
4. Simulation Experiments and Results for Unmanned Aerial Vehicle Path Planning
4.1 Experimental Setup
- Hardware/Software: Windows 11, Intel Core i5-13400 @ 2.50GHz, 16GB RAM, MATLAB R2023a.
- UAV Environment: Flight area: x∈[0,150]x∈[0,150] km, y∈[0,100]y∈[0,100] km, z∈[0,2.2]z∈[0,2.2] km. Start: S=(10,90,1.1146)S=(10,90,1.1146) km, Goal: G=(130,10,1.3666)G=(130,10,1.3666) km.
- Path Parameters: Navigation points N=5N=5, Weights w1=0.7w1=0.7, w2=0.2w2=0.2, w3=0.1w3=0.1, Penalty p=104p=104.
- Obstacle Scenarios:
- Scenario 1: 6 obstacles.
- Scenario 2: 10 obstacles (includes all obstacles from Scenario 1 plus 4 more).
- Algorithms Compared: IMTBO, Standard MTBO, Dung Beetle Optimizer (DBO), Whale Optimization Algorithm (WOA), Grey Wolf Optimizer (GWO).
- Algorithm Settings: Population size Npop=30Npop=30, Max iterations T=200T=200. Each algorithm run independently 20 times per scenario. IMTBO parameters: α=0.5α=0.5, k=0.75k=0.75, S=0.7S=0.7, a=1.6a=1.6, σ=0.1⋅rangeσ=0.1⋅range. Other algorithms used standard literature parameters.
- Evaluation Metrics: Best fitness value LbestLbest found, Average fitness LavgLavg over 20 runs, Standard deviation σLσL of fitness over 20 runs (measure of stability), Convergence speed (iterations to find good solution), Visual path quality.
Table 3: Obstacle Information for Unmanned Aerial Vehicle Path Planning Scenarios
| Scenario | Obstacle ID | Center (x, y) [km] | Radius [km] |
|---|---|---|---|
| 1 (6 Obs) | 1 | (10, 60) | 5 |
| 2 | (20, 35) | 4 | |
| 3 | (40, 50) | 6 | |
| 4 | (65, 30) | 5 | |
| 5 | (80, 60) | 8 | |
| 6 | (100, 30) | 4 | |
| 2 (10 Obs) | 1-6 | (Same as Scenario 1) | (Same) |
| 7 | (30, 45) | 7 | |
| 8 | (50, 20) | 5 | |
| 9 | (70, 70) | 6 | |
| 10 | (90, 10) | 4 |
4.2 Results and Analysis
- Path Visualization: Analysis of the planned paths revealed that regions with dense obstacles (e.g., around (10, 60), (65, 30), (80, 60) km) presented significant local optima challenges. While algorithms like MTBO, GWO, and DBO often exhibited stagnation or sub-optimal path segments near these clusters, WOA and IMTBO demonstrated superior capability in navigating through complex obstacle fields. IMTBO consistently produced the shortest and visually smoothest paths in both scenarios.
- Convergence Speed: IMTBO exhibited significantly faster convergence compared to all other algorithms, including WOA. In Scenario 1, IMTBO typically found its best solution within approximately 30 iterations. The convergence curves clearly showed IMTBO achieving lower fitness values much earlier and maintaining a steeper descent rate.
- Solution Quality and Stability: Statistical results over 20 independent runs for each algorithm and scenario are summarized below. Fitness LL represents the penalized objective function value (lower is better).
*Table 4: Performance Comparison of Path Planning Algorithms for Unmanned Aerial Vehicle (Scenario 1 – 6 Obstacles)*
| Algorithm | Best Fitness LbestLbest [km] | Average Fitness LavgLavg [km] | Std. Dev. σLσL [km] |
|---|---|---|---|
| IMTBO | 101.32 | 103.03 | 2.05 |
| MTBO | 101.37 | 108.45 | 14.68 |
| DBO | 101.90 | 114.29 | 6.56 |
| GWO | 108.74 | 121.62 | 14.33 |
| WOA | 103.90 | 111.32 | 5.29 |
*Table 5: Performance Comparison of Path Planning Algorithms for Unmanned Aerial Vehicle (Scenario 2 – 10 Obstacles)*
| Algorithm | Best Fitness LbestLbest [km] | Average Fitness LavgLavg [km] | Std. Dev. σLσL [km] |
|---|---|---|---|
| IMTBO | 101.32 | 103.12 | 2.58 |
| MTBO | 101.50 | 122.53 | 20.19 |
| DBO | 105.14 | 115.68 | 8.69 |
| GWO | 105.88 | 130.41 | 23.08 |
| WOA | 103.19 | 111.39 | 6.18 |
Key Observations from Results:
- Superior Best Fitness: IMTBO consistently achieved the lowest (best) LbestLbest value in both scenarios, indicating its superior ability to find the shortest, safest, and smoothest paths for the Unmanned Aerial Vehicle.
- Significantly Better Average Fitness: The average fitness LavgLavg of IMTBO was markedly lower than all competitors (MTBO, DBO, GWO, WOA) in both scenarios. This demonstrates its robustness and consistency across multiple runs.
- Exceptional Stability (Low Std. Dev.): IMTBO exhibited the smallest standard deviation σLσL by a significant margin in both scenarios. This low variance confirms that IMTBO is highly stable and reliable, producing consistently good path planning solutions for the Unmanned Aerial Vehicle regardless of initial conditions. The stability of MTBO, GWO, and DBO was notably poor, especially in the more complex Scenario 2.
- Robustness to Complexity: While the performance of MTBO, DBO, and GWO deteriorated substantially when moving from Scenario 1 (6 obstacles) to Scenario 2 (10 obstacles), IMTBO maintained its high performance level. The increase in its average fitness and standard deviation was minimal compared to the drastic increases seen in other algorithms, highlighting IMTBO’s robustness in handling increasingly complex Unmanned Aerial Vehicle environments.
- Fast Convergence: As noted visually from convergence curves, IMTBO converged significantly faster than all other algorithms, reaching near-optimal solutions in a fraction of the iterations. This rapid convergence is critical for time-sensitive Unmanned Aerial Vehicle applications.
5. Conclusion
This work presented the Improved Mountaineering Team-Based Optimization (IMTBO) algorithm for solving the complex 3D path planning problem of Unmanned Aerial Vehicles. IMTBO strategically enhances the standard MTBO algorithm through four key innovations applied to the Unmanned Aerial Vehicle path encoding and optimization process:
- Enhanced Initialization: Combining Tent Chaotic Mapping and Refraction Reverse Learning (RRL) significantly improved the initial population diversity and coverage of the high-dimensional search space, leading to better exploration potential crucial for navigating complex Unmanned Aerial Vehicle environments.
- Balanced Exploration-Exploitation in Disaster Phase: Replacing the original Disaster Threat update with the Sine Cosine Strategy (SCS) provided a dynamic mechanism for balancing global search (exploration) and local refinement (exploitation), effectively helping the algorithm escape local optima traps common in obstacle-rich Unmanned Aerial Vehicle path planning.
- Refined Local Search in Member Update: Substituting random re-initialization with Gaussian Mutation introduced small, localized perturbations, enhancing the algorithm’s ability to fine-tune solutions (exploitation) during later stages, leading to smoother and shorter Unmanned Aerial Vehicle paths.
- Effective Constraint Handling: The integration of constraint violation penalties into the fitness function allowed IMTBO to effectively navigate the Unmanned Aerial Vehicle path around obstacles while respecting altitude and boundary limits.
Comprehensive simulation experiments, comparing IMTBO against MTBO, DBO, WOA, and GWO across two scenarios with different obstacle densities, unequivocally demonstrated the superiority of IMTBO for Unmanned Aerial Vehicle path planning:
- IMTBO consistently found paths with the lowest cost (shortest length, controlled height variation, high smoothness).
- IMTBO exhibited significantly faster convergence, finding high-quality solutions in fewer iterations.
- IMTBO showed remarkable stability and robustness, evidenced by the lowest standard deviation in solution quality across multiple runs, especially in the more complex environment. Its performance degradation with increased obstacles was minimal compared to other algorithms.
The proposed IMTBO algorithm provides a highly effective, efficient, and robust solution for generating optimal and feasible 3D flight paths for Unmanned Aerial Vehicles operating in complex, obstacle-laden environments. Future work involves extending IMTBO to dynamic path planning scenarios and multi-Unmanned Aerial Vehicle cooperative path planning.