## 3 Relative Kinematic-Equation Building for UAV-UGV Formations  
In this study, we investigate the UAV-UGV formation-maintenance control problem for cooperative combat mission requirements. Considering the actual motion state of a UGV as the input perturbation of the formation-maintenance controller, we assume that the control system of the UAV is closed-loop stable, i.e., the directions of the UAV velocity command, trajectory-inclination command, and trajectory-declination command can be stably followed. The three channels were set in the following first-order system [31]:  
\[\left\{ \begin{array}{l}\dot{V}_{f}=-\frac{1}{\tau_{off}}\Big{(}V_{f}-V_{fc} \Big{)}\\ \dot{\theta}_{f}=-\frac{1}{\tau_{off}}\Big{(}\theta_{f}-\theta_{fc}\Big{)}\\ \dot{\psi}_{ref}=-\frac{1}{\tau_{off}}\Big{(}\psi_{ref}-\psi_{ref}\Big{)}\end{array} \right. \tag{1}\]  
where \(\tau_{off}\) denotes the inertia time constant of the velocity control channel of the UAV, \(\tau_{off}\) denotes the inertia time constant of the track-inclination control channel of the UAV, \(\tau_{\psi_{ref}}\) denotes the inertia time constant of the track-declination control channel of the UAV, \(V_{f}\) is the actual velocity magnitude of the UAV, \(\theta_{f}\) is the actual inclination angle of the UAV, \(\psi_{ref}\) is the actual declination angle of the UAV, \(V_{fc}\) is the commanded velocity magnitude of the UAV, \(\theta_{fc}\) is the commanded inclination angle of the UAV, and \(\psi_{ref}\) is the commanded declination angle of the UAV. To study the relative motion between the UGV and UAV, the following coordinate systems are first defined:  
(1) Inertial coordinate system \(O_{1}X_{1}Y_{1}Z_{1}\): The origin \(O_{1}\) is located at a fixed point on the ground, the \(O_{1}X_{1}\) axis points to the target, the \(O_{1}Y_{1}\) axis is vertically upward, and the \(O_{1}Z_{1}\) axis forms a right-handed coordinate system with the first two axes. (2) Relative coordinate system \(O_{r}X_{r}Y_{r}Z_{r}\): The origin \(O_{r}\) is located at the center of mass of the UAV, the \(O_{r}X_{r}\) axis points in the direction of the UGV velocity, the \(O_{r}Y_{r}\) axis is in the vertical plane perpendicular to the \(O_{r}X_{r}\) axis, and the \(O_{r}Z_{r}\) axis forms a right-handed coordinate system with the first two axes. (3) Vehicle-body coordinate system \(O_{2}X_{2}Y_{2}Z_{2}\): The origin \(O_{2}\) is located at the center of mass of the UAV, the \(O_{2}X_{2}\) axis points in the direction of the UAV velocity, the \(O_{2}Y_{2}\) axis is in the vertical plane perpendicular to the \(O_{2}X_{2}\) axis, and the \(O_{2}Z_{2}\) axis forms a right-handed coordinate system with the first two axes. In the relative coordinate system, according to [PERSON] theory, the motions of the UGV and UAV are related as follows:  
\[\frac{d\mathbf{r}}{dt}=\mathbf{V}_{fr}-\mathbf{V}_{lr}=\mathbf{V}_{r}+\mathbf{\omega}\times\mathbf{r} \tag{2}\]  
where \(\mathbf{r}\) is the relative radius vector between the UGV and UAV; \(\mathbf{V}_{fr}\) and \(\mathbf{V}_{lr}\) are the velocity vectors of the UAV and UGV in the relative coordinate system, respectively; \(\mathbf{V}_{r}\) is the derivative of the radius vector \(\mathbf{r}\) in the relative coordinate system; \(\mathbf{\omega}\) is the angular velocity of rotation of the relative coordinate system in the inertial coordinate system. To obtain the absolute velocity of the UGV and UAV in the relative coordinate system, the following transformation can be performed:  
\[\left\{ \begin{array}{l}\mathbf{V}_{ft}=\Phi_{1}^{r}\Phi_{2}^{l}V_{f2}\\ \mathbf{V}_{lr}=\mathbf{V}_{l2}\end{array} \right. \tag{3}\]  
where \(\Phi_{1}^{r}\) is the conversion matrix from the inertial to the relative coordinate system; \(\Phi_{2}^{l}\) is the conversion matrix from the UAV body coordinate system to the inertial coordinate system; \(\mathbf{V}_{f2}\) is the velocity vector under the UAV body coordinate system; \(\mathbf{V}_{l2}\) is the velocity vector under the UGV body coordinate system.