Joint Three-Dimensional Path Planning for Multiple Unmanned Aerial Vehicles in Static Environments Using an Improved Whale Optimization Algorithm

Coordinated multi-unmanned aerial vehicles (UAVs) planning requires the joint consideration of spatial paths, flight speeds, arrival times, and separation constraints. This study develops an improved whale optimization algorithm (IWOA) for offline path–speed planning in known static environments. The method combines continuous spherical-increment encoding with deterministic initialization, nonlinear adaptive weighting, population migration, and a heavy-tailed candidate. Among eight algorithms, IWOA achieved the best mean rank of 1.38 on 29 2017 IEEE Congress on Evolutionary Computation (CEC2017) functions and a mean rank of 1.25 across four repeated single-UAV scenarios. Eight algorithms were also evaluated in four 230-dimensional, five-UAV scenarios using 30 runs, a population of 30, and 500 iterations, for 960 runs in total. The penalty-only IWOA obtained the lowest median final fitness in three scenarios, but no solution passed the post hoc feasibility screen. We therefore introduced a feasibility-priority variant with corrected angular and geometric checks, normalized violation magnitudes, continuous-time separation checks, and Deb-style ordering. In a separate 240-run comparison, Improved Whale Optimization Algorithm with Feasibility-Priority (IWOA-FP) and Whale Optimization Algorithm with Feasibility-Priority (WOA-FP) each returned 120/120 modeled-feasible terminal incumbents from a shared, prevalidated feasible warm start. Conditional objective comparisons favored IWOA-FP in Map 4, Complexity Setting 1 (M4-C1) and Map 4, Complexity Setting 2 (M4-C2) and WOA-FP in Map 3, Complexity Setting 2 (M3-C2), with neither method favored in Map 3, Complexity Setting 1 (M3-C1). These results show that IWOA is competitive under fixed-generation testing and that explicit feasibility-priority selection can preserve and optimize a feasible plan. Validation remains limited to the tested static model, and neither measures unseeded feasibility discovery nor certifies flight safety.