<![CDATA[Accurate trajectory tracking of differential-drive mobile robots is strongly influenced by proportional–integral–derivative (PID) controller parameters, while a single fixed parameter set may exhibit different performance across trajectories with distinct geometric characteristics. This study investigates the multi-trajectory performance and run-to-run consistency of Grey Wolf Optimizer (GWO)-based PID tuning for a differential-drive mobile robot. A kinematic robot model with linear and angular-velocity PID control is evaluated on four reference trajectories: circle, lemniscate, square, and S-curve. A systematically tuned fixed PID controller is employed as the common baseline, while GWO independently optimizes six PID gains for each trajectory using an objective function combining the integral of time-weighted absolute error (ITAE) and control effort. To account for the stochastic nature of GWO, 30 independent optimization runs are performed for each trajectory and performance is assessed using root mean square error (RMSE), ITAE, standard deviation, median, interquartile range, and coefficient of variation. The simulation results show that GWO-PID reduces the mean RMSE from 0.07767 to 0.04979 m for the circle, from 0.08113 to 0.06393 m for the lemniscate, and from 0.09047 to 0.05095 m for the S-curve, corresponding to improvements of 35.90%, 21.20%, and 43.68%, respectively. The square trajectory exhibits an increase in mean RMSE from 0.07728 to 0.08154 m, indicating that optimization does not provide uniform improvement across all trajectory geometries. Run-to-run analysis further reveals substantial differences in optimization consistency with RMSE coefficients of variation of 54.17% for the circle, 85.33% for the lemniscate, and 12.03% for the square, and 0.75% for S-curve. These findings indicate that the effectiveness and repeatability of GWO-based PID tuning are trajectory-dependent, highlighting the importance of multi-run statistical evaluation when assessing metaheuristic controller tuning for mobile robot trajectory tracking.]]>
