## 5 Simulation Analysis  
In this section, we describe the numerical simulations performed for the UAV-UGV formation-maintenance control system consisting of a UGV with a given motion state, a UAV with a first-order stability control system, and an optimal maintenance controller. The inertial time constants of the three channels of the UGV and UAV are assumed to be \(\tau_{\mathit{off}}=3\) s, \(\tau_{\mathit{off}}=1\) s, and \(\tau_{\mathit{\bar{p}_{\mathit{off}}}}=1\) s. The initial positions of the UAV and UGV in the inertial coordinate system are chosen as \((-5,20,10)\) m and \((0,0,0)\) m, respectively. It is assumed that the UAV-UGV formation moves for 150 s, and the simulation step size is taken to be 0.01 s. Under the action of Equation (43), the entire motion of the UAV is adjusted and maintained from the initial position to the desired position at \((-5,20,10)\) m. In all simulations, the required control system power coefficient matrix in Equation (34) appears as the matrices shown in Equation (47). Considering the feasible motion envelope of the UAV, limits are set for the control amount of the formation controller and commanded motion state of the UAV, as shown in Equation (52).  
\[\mathbf{Q}_{y}=\left[\begin{array}{ccc}4&0&0\\ 0&6&0\\ 0&0&6\end{array}\right],\mathbf{R}=\left[\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right],\mathbf{R}_{1}=\left[\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right] \tag{47}\]  
To verify the effectiveness of the proposed UGV-UAV formation-maintenance controller, we set two cases in which the UGV adopts different motion modes. The motion of the UGV is set as in Equations (48) to (51). The simulation results are shown in Figures 1 and 2.  
case 1:  
\[V_{l}(t)=\left\{\begin{array}{l}2+\sin(0.1t)\;\mathrm{m/s},0\leq t<50\; \mathrm{s}\\ 2\;\mathrm{m/s},50\leq t<100\;\mathrm{s}\\ 2+\sin(0.1t)\;\mathrm{m/s},100\leq t<150\;\mathrm{s}.\end{array}\right. \tag{48}\]  
\[\left\{\begin{array}{l}\theta_{l}(t)=0\;\mathrm{rad}\\ \psi_{vl}(t)=0.5\cos(0.1t+0.7728)\;\mathrm{rad}\end{array}\right. \tag{49}\]  
case 2:  
\[V_{l}(t)=\left\{\begin{array}{l}1.5\;\mathrm{m/s},0\leq t<50\;\mathrm{s}\\ 2+\sin(0.1t)\;\mathrm{m/s},50\leq t<100\;\mathrm{s}\\ 1.5\;\mathrm{m/s},100\leq t<150\;\mathrm{s}\end{array}\right. \tag{50}\]  
\[\left\{\begin{array}{l}\theta_{l}(t)=0\;\mathrm{rad}\\ \psi_{vl}(t)=0.2\sin(0.15t+0.5523)\;\mathrm{rad}\end{array}\right. \tag{51}\]  
\[\left\{\begin{array}{l}1\;\mathrm{m/s}\leq V_{fc}\leq 3\;\mathrm{m/s}\\ -15^{\circ}\leq\theta_{fc}\leq 25^{\circ}\\ -80^{\circ}\leq\psi_{7 fc}\leq 80^{\circ}\end{array}\right. \tag{52}\]  
It can be seen from Figures 1 and 2 that under the control command in Equation (46), the UGV can guide the UAV to fly at a fixed altitude in accordance with the specified motion, which results in the maintenance of a fixed formation in the three-dimensional plane. In addition, the velocity and declination angle commands of the UVA vary slightly and reasonably without the jitter vibration phenomenon, which satisfies the constraints of the laws of physics. Moreover, the motion of the UGV perturbs the UAV-UGV formation-maintenance control. As the UGV changes its motion form at 50 and 100 s, the motion trajectory of the UAV fluctuates at the corresponding moments. Nevertheless, it can be quickly stabilized at the desired motion state using a formation-maintenance controller. In case 1, the maximum values of the motion state errors of the UAV in the \(X_{r}\), \(Y_{r}\), and \(Z_{r}\) directions are 1.2 m, 0 m, and 1.8 m, respectively. In Case 2, the maximum values of the motion state errors of the UAV in the \(X_{r}\), \(Y_{r}\), and \(Z_{r}\) directions are 0.6 m, 0 m, and 0.6 m, respectively.  
Figure 1: Simulation results of case 1. (**a**) Three-dimensional motion trajectory of UGV and UAV; (**b**) velocity of UAV; (**c**) deflection angle of UAV; (**d**) motion trajectory in \(X_{r}\)-axis direction; (**e**) motion trajectory in \(Y_{r}\)-axis direction; (**f**) motion trajectory in \(Z_{r}\)-axis direction.