## 5 Simulation Results  
As shown in Figure 4, to verify the effectiveness of the proposed fault-tolerant control algorithm, we employed the QBall-X4 UAV platform for UAV data acquisition and flight verification, and the platform could effectively achieve the injection of real actuator fault data. In order to realistically analyze the control performance of the proposed fault-tolerant control algorithm, we collected real flight data in the experimental phase, which mainly included the state data \(\big{[}\phi,\psi,\theta,\dot{\phi},\dot{\theta},\dot{\psi}\big{]}\), control signal \(\big{[}u_{1},u_{2},u_{3},u_{4}\big{]}\), and the desired signal \(\left[\phi_{ref},\psi_{ref},\beta_{ref}\right]\) of the remote control in manual control mode. In this paper, we mainly present the performance curve of the attitude controller to verify and demonstrate the effectiveness of the designed algorithm, where the selected roller subsystem can be expressed as follows:  
\[\left\{ \begin{array}{l}\ddot{\phi}=\left(\dot{\phi}\dot{\psi}\frac{I_{yy}-I_{ zz}}{I_{xx}}+\xi_{1}(x)\right)+\frac{l}{I_{xx}}u_{2}+\xi_{2}(x)u_{2}+d_{w}\\ y=\dot{\phi}\end{array} \right.. \tag{50}\]  
where \(f_{0}=\dot{\phi}\dot{\psi}\frac{I_{yy}-I_{zz}}{I_{xx}}\), \(g_{0}=\frac{l}{I_{xx}}\), \(J_{xx}=0.01194\) kg\(\cdot\)m\({}^{2}\), \(J_{yy}=0.01194\) kg\(\cdot\)m\({}^{2}\), \(J_{zz}=0.02181\) kg\(\cdot\)m\({}^{2}\), \(l=0.2\) m, and the parameters \(\xi_{1}\) and \(\xi_{2}\) are the parts of the system for which parameter fusion estimation is required. In ST-RBFNN, the state variables were used as the input for model training, and the collected data were then used as parameters and for subsequent calculations. The raw dataset was a combination of the standard flight data and the fault data from a fault injection test UAV. In this study, each set of experimental data was collected at a sampling frequency of 1 ms, and each set contained 60,000 consecutive time series of data, with the training data and test data divided in a 5:1 ratio. The neural network was then iteratively trained according to the weight update rule of the novel ST-RBFNN designed in Section 3. Finally, the trained network model was embedded in the controller in the experiments. As shown in Figure 5, given a set of 500 sample data for validation and comparison, the average mean squared error of the proposed RBFNN based on the new adaptive kernel was found to be better than that of the conventional RBFNN. Figure 4: Schematic diagram of the experimental system. Figure 5: Testing MSE curves of ST\(-\)RBF and conventional RBF. According to Figures 6 and 7, ST-RBFNN tracked the predicted sample data more accurately than the conventional RBFNN, with a tracking error close to zero. In the validation phase of the effectiveness of the fault-tolerant controller, the pitch and roll systems selected in this study were compared with the experimental results. To validate the effectiveness of the proposed fault-tolerant controller, a multi-rotor UAV with modeling uncertainties was selected in this study. Additionally, three flight simulation scenarios were considered for the qualitative analysis of the control performance of the controller. **Scenario 1**: Testing the flight effects of an active disturbance rejection, fault-tolerant controller based on a spatio-temporal RBF predictive neural network in the absence of external wind gust disturbances and actuator failures. **Scenario 2**: Numerical simulation of a horizontal wind gust disturbance of 3-9 m/s. The controller's performance was analyzed on the basis of the effect of the aircraft in a sustained wind disturbance scenario. The performance of the controller was analyzed on the basis of the effects of the aircraft in a wind disturbance situation that lasts for a period of time.