## 5 Simulation Results  
To validate the proposed framework, we conduct simulation experiments for a quadrotor with 3D wind disturbances. We utilize Visual Studio Code as the software environment and Python as the programming language. We perform the simulation with the proposed method by referencing the code from the paper [12]. The wind speed varies within a range of \(-5\) m/s to \(5\) m/s along the x-axis, and \(-2.5\) m/s to \(2.5\) m/s along the z-axis. The wind condition randomly switches to the next one every \(2\) s. The parameters pertaining to the controller and the quadrotor platform are given in Table 1. The maximum motor speed is set to \(1000\) rad/s and the maximum quadrotor thrust is set to \(125\) N.  
To verify the effectiveness of the meta-learning method on the control matrix, we utilize INDI control combined with the learned matrix for a quadrotor against wind disturbances. We can see in Figure 3 that as the system encounters more environments the controller performs better. This can be observed from the decreasing position and velocity errors along the x-axis over time. We select three random situations and compare the control effectiveness of PID and INDI methods with the control matrix after meta-learning, respectively. We compare the Root Mean Square Error (RMSE), which represents the deviation of the quadrotor from the hover point under wind disturbances in these different situations. From Table 2 we can see that the proposed meta-learning-based INDI control method significantly outperforms the baseline INDI and it also performs considerably better than the meta-learning-based PID. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** \\ \hline \(k_{x}\) & \(\begin{bmatrix}6&0&0\\ 0&6&0\\ 0&0&3\\ 4&0&0\\ 0&4&0\\ 0&0&3\\ \end{bmatrix}\) \\ \(g\) & \(\begin{bmatrix}9.81\\ 2.4\\ 0.0036&0\\ 0&0.0036\\ 0&0&0.0036\\ \end{bmatrix}\) \\ \(J\) (kg\(\cdot\) m\({}^{2}\)) & \(\begin{bmatrix}0.0036&0&0\\ 0&0.0036&0\\ 0&0&0.0036\\ \end{bmatrix}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Controller and quadrotor platform parameters. Figure 3: Tracking errors along the x-axis. The first subfigure shows the actual position and the desired position along the x-axis. The position error is decreasing as the system encounters more situations. The second subfigure shows that in comparison with the wind disturbances, the size of position error is quite small. The third subfigure shows the actual velocity and the desired velocity along the x-axis. The velocity error is decreasing similarly. We compare the disturbance rejection capabilities of the proposed method and the OMAC method for quadrotors under a specific random wind condition to maintain a hovering state. The hover point is located at (0, 0, 0). As shown in Figures 4 and 5, the flight trajectory of the quadrotor with the proposed method is closer to the desired hover point, indicating a better control performance. This is quantitatively reflected in the RMSE values, with the meta-learning-based INDI method yielding an RMSE of 0.057 m, significantly lower than 0.139 m observed with the OMAC method. The quadrotor operated by the proposed method exhibits a smaller position error diminishing progressively over time, as depicted in Figure 6, validating the effectiveness of the approach in minimizing discrepancies from the desired position. These results demonstrate the superiority of the proposed meta-learning-based INDI control method in enhancing the adaptability of quadrotors in continuously changing disturbance environments. For one random situation the ACE values on the x-axis and z-axis are as shown in Figure 7. We consider the mass-asymmetric scenario first. Vertical lines in the figure indicate that the wind condition is switched. The performance of the meta-learning-based INDI method and the meta-learning-based PID method progressively improve in comparison with baseline, as they can benefit from prior experiences when they encounter more environments.