## 4 Results
### Undisturbed Trajectory Tracking  
To assess the trajectory tracking capabilities of our proposed hybrid control strategy, we assume a continuous and differentiable desired tracking trajectory. The chosen trajectory is defined as \(x_{d}=8 sin(0.1t),y_{d}=8 sin\left(0.1t+\frac{1}{2}\pi\right),\) and \(z_{d}=-8+0.2t\), with a constant desired yaw angle \(\psi_{d}\) of zero. Initially, the body is positioned at \(P^{E}=\left[0\,0-8\right]^{T}\), while the velocities \(V^{E}=\Theta^{E}=\Omega^{B}=0^{3\times 1}\). We conduct numerical simulations in the absence of disturbances, utilizing the control parameters listed in Table 2. Figure 4 illustrates the performance of our proposed control method, which exhibits slight deviations compared to other control methods [32; 33; 34]. In comparison to other control methods, the proposed approach demonstrates superior trajectory tracking performance. While the traditional backstepping control strategy results in a shorter time for the unmanned aerial vehicle to reach the desired trajectory compared to the bioinspired backstepping control strategy, Figure 5 shows that the velocity commands generated by the former exhibit abrupt changes. On the other hand, the bioinspired hybrid control method produces smoother velocity commands without any jumps. The abrupt change in velocity commands is a crucial issue for practical applications in unmanned aerial vehicles because it demands the immediate attainment of unrealistic velocities and torques when there is a significant initial error, which is not achievable with existing actuators. Furthermore, this issue may pose a risk of crashes during flight. Therefore, the bioinspired backstepping control strategy effectively addresses this problem. Furthermore, from Figure 5(a), it can be observed that the traditional sliding mode control strategy suffers from control signal chattering, which imposes a significant burden on the onboard actuators. In contrast, the proposed bioinspired sliding mode controller effectively avoids this issue and provides smooth torque control inputs. The combination of saturation function and sliding mode control is a common approach that can effectively mitigate the chattering problem associated with some traditional sliding mode control methods. However, when the tracking error \(e_{\Omega}\) is outside the interval defined by \(B_{\Omega}\) and \(D_{\Omega}\), the control signal may still exhibit chattering. When the unmanned aerial vehicle operates in a complex environment, it is challenging to tune the control parameters of the saturation function to achieve satisfactory performance under varying flight conditions. As shown in Figure 5(b), although the chattering problem is partially addressed, some chattering still persists. Hence, the saturation function alone cannot fully resolve the chattering issue in traditional sliding mode control, while the bioinspired sliding mode control strategy, as observed from Figure 6, demonstrates good performance in handling chattering. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Value** \\ \hline \(k_{t}\) & 0.6 \\ \(G_{\Omega}\) & \(diag(1,\,1,\,1)\) \\ \(A_{\rm P}\) & \(diag(-0.4,-0.4,-0.4)\) \\ \(B_{\rm P}\) & \(diag(0.8,\,0.8,\,0.8)\) \\ \(D_{\rm P}\) & \(diag(-0.8,\,-0.8,\,-0.8)\) \\ \(A_{\rm V}\) & \(diag(-0.25,-0.25,-0.25)\) \\ \(B_{\rm V}\) & \(diag(1,\,1,\,1)\) \\ \(D_{\rm V}\) & \(diag(-1,-1,-1)\) \\ \(A_{\rm S}\) & \(diag(-0.825,-0.825,-0.825)\) \\ \(B_{\rm S}\) & \(diag(1.8,1.8,\,1.8)\) \\ \(D_{\rm S}\) & \(diag(-1.8,\,-1.8,\,-1.8)\) \\ \(K_{\rm P}\) & \(diag(1.48,\,1.48,\,1.2)\) \\ \(K_{V}\) & \(diag(6,\,6,\,4.8)\) \\ \(a_{1}\) & \(diag(0.1,\,0.1,\,0.1)\) \\ \(a_{2}\) & \(diag(0.01\,,\,0.01,\,0.01)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Controller parameters. Figure 4: Trajectory tracking performance. CB, traditional backstepping [12]; SMC, traditional sliding mode [33]; SAT, sliding mode with saturation [19]; BB, bioinspired backstepping; BSMC, bioinspired sliding mode.