## 2 Problem Analysis and Formulation
### The Motion Model of the UCAV  
Firstly, the engagement geometry relationship is established in three-dimensional space, as shown in Figure 3.  
The following coordinate systems are involved in Figure 3, \(Ox_{g}y_{g}z_{g}\), \(O_{u}x_{gu}y_{gu}z_{gu}\), \(O_{u}x_{hu}y_{hu}z_{hu}\), \(O_{t}x_{gt}y_{gt}z_{gt}\), and \(O_{t}x_{ht}y_{ht}z_{ht}\), which represent the geographic coordinate system, the concomitant inertial coordinate system of the UCAV, the trajectory coordinate system of the UCAV, the concomitant inertial coordinate system of the target, and the trajectory coordinate system of the target, respectively. \(\varepsilon\) and \(\beta\) are the inclination angle and the deflection angle of line-of-sight of the UCAV, respectively, and \(D\) is the distance between the UCAV and target.  
The UCAV is modeled as a three-degree-of-freedom, point-mass model in three-dimensional, which can be described by \(\dot{x}_{u}=f_{u}(x_{u},\ \mathbf{u}_{u})\) and \(\mathbf{x}_{u}(t_{0})=\mathbf{x}_{u0}\), and details are as following:  
\[\left\{\begin{array}{l}\dot{x}_{u}=v_{u}\cos\theta_{u}\cos\phi_{u}\\ \dot{y}_{u}=v_{u}\sin\theta_{u}\\ \dot{z}_{u}=v_{u}\cos\theta_{u}\sin\phi_{u}\\ \dot{v}_{u}=g(n_{xc}-\sin\theta_{u})\\ \dot{\theta}_{u}=\frac{\beta_{v}}{v_{u}}(n_{yc}-\cos\theta_{u})\\ \dot{\phi}_{u}=\frac{\beta_{vc}}{v_{u}\cos\theta_{u}}\end{array}\right. \tag{1}\]  
where the state vector \(\mathbf{x}_{u}\) refers to \(\mathbf{x}_{u}=[x_{u},y_{u},z_{u},v_{u},\theta_{u},\phi_{u}]^{\rm T}\) and \(\mathbf{x}_{u0}\) to the initial state vector. \(x_{u}\) and \(y_{u}\) refer to the horizontal coordinates and \(y_{u}\) to the altitude of the UCAV in the geographic coordinate system. \(v_{u}\) is the velocity of the UCAV and \(g\) is the gravitational acceleration. \(\theta_{u}\) and \(\phi_{u}\) are the flight path angle and the heading angle of the UCAV, respectively. The control vector \(\mathbf{u}_{u}\) refers to \(\mathbf{u}_{u}=[n_{xu},\ \mathbf{n}_{yu},\ n_{zu}]^{\rm T}\), where \(n_{xu}\), \(n_{yu}\), and \(n_{zu}\) are the required control overload components of the corresponding axis in the trajectory coordinate system for the UCAV. \(\mathbf{u}_{u}\) is given by the designed attack guidance system.  
The control overload constraint model of the UCAV is designed as  
\[\left\{\begin{array}{l}n_{yc}=\frac{\min(n_{max},n_{u})}{n_{u}}n_{yu}\\ n_{zc}=\frac{\min(n_{max},n_{z})}{n_{zu}}n_{zu}\end{array}\right. \tag{2}\]  
Figure 3: Engagement geometry relationship between the UCAV and the target.  
where \(n_{u}=\sqrt{n_{zu}^{2}+n_{yu}^{2}}\) and \(n_{\text{max}}\) is the maximum available overload of the UCAV in the normal direction. \(n_{xc}\), \(n_{yc}\) and \(n_{zc}\) are the actual control overloads of the UCAV. The target motion model similar to that of the UCAV is described by \(\hat{\mathbf{x}}_{t}=f_{t}(\mathbf{x}_{t},\ \mathbf{u}_{t})\) and \(\mathbf{x}_{t}(t_{0})=\mathbf{x}_{t0}\), where \(\mathbf{x}_{t}=[x_{t},y_{t},z_{t},v_{t},\theta_{t},\phi_{t}]^{\text{T}}\) and \(\mathbf{u}_{t}=[n_{xt},\ \mathbf{u}_{yt},\ n_{zt}]^{\text{T}}\). The corresponding parameters definition in the motion model of the target is the same as that in the UCAV.