## 2 Dynamic Models and Constraints of Re-Entry Glide Vehicle  
### Dimensionless Dynamic Model in the Flight Range Domain  
In order to facilitate the trajectory generation of the vehicle, the dynamic model of the vehicle is usually dimensionless in the time domain. The dimensionless dynamic equation is \[\left\{\begin{array}{l}\dot{r}=v\sin\theta\\ \dot{\phi}=v\cos\theta\sin\psi/r\cos\varphi\\ \dot{\varphi}=v\cos\theta\cos\psi/r\\ \dot{v}=-D-\sin\theta/r^{2}\\ \dot{\theta}=L\cos\beta/v+v\cos\theta/r+\cos\theta/\left(vr^{2}\right)+C_{ \theta}+\widetilde{C}_{\theta}\\ \dot{\psi}=L\sin\beta/(v\cos\theta)+v\cos\theta\sin\psi\tan\varphi/r+C_{ \phi}+\widetilde{C}_{\psi}\end{array}\right. \tag{1}\]  
where, \(r\) represents the dimensionless geocentric distance of the vehicle, \(\phi\), \(\varphi\) represent the longitude and latitude coordinates, \(v\) represents the dimensionless speed, \(\theta\) represents the flight path angle, \(\psi\) represents the flight heading angle of the target, \(L\) and \(D\) represent the dimensionless lift and drag of the vehicle, \(C_{\theta}\), \(\widetilde{C}_{\theta}\) and \(C_{\phi}\), \(\widetilde{C}_{\phi}\) represent the coriolis acceleration item and the involved acceleration item caused by the earth's rotation.  
\[\left\{\begin{array}{l}L=\left(\rho(v\cdot v_{c})^{2}SC_{L}\right)/2 mg_{0}\\ D=\left(\rho(v\cdot v_{c})^{2}SC_{D}\right)/2 mg_{0}\end{array}\right. \tag{2}\]  
where, \(C_{L}\) represents the lift coefficient, \(C_{D}\) represents the drag coefficient, \(S\) represents the reference area of the vehicle, \(m\) represents the mass of the vehicle, and \(g_{0}\) represents the gravity acceleration at zero altitude, \(v_{c}=\sqrt{g_{0}R_{e}}\) represents the dimensional velocity where \(R_{e}\) is the radius of earth, \(\rho\) is the atmosphere density. The dimensionless operation of other quantities can be referred to in the literature [28].  
Define the initial range-to-go of the re-entry glide vehicle as \(S_{f0}\).  
\[S_{f0}=\arccos\left(\cos\varphi_{f}\cos\varphi_{0}\cos(\phi_{f}-\phi_{0})+\sin \varphi_{f}\sin\varphi_{0}\right) \tag{3}\]  
where, \(\phi_{0},\varphi_{0}\) represent the longitude and latitude coordinates of the initial point, and \(\phi_{f},\varphi_{f}\) represent the longitude and latitude coordinates of the target point.  
Then the speed of the vehicle  
Operating Equation (1) with Equation (4), the dynamic model of the vehicle in the flight range domain can be expressed as  
\[\left\{\begin{array}{l}dr/ds=-r\tan\theta/\cos(\psi-\psi_{p})\\ d\phi/ds=-\sin\psi\big{/}\big{(}\cos\varphi\cos(\psi-\psi_{p})\big{)}\\ d\varphi/ds=-\cos\psi/\cos(\psi-\psi_{p})\\ dv/ds=Dr/v\cos\theta\cos(\psi-\psi_{p})+\tan\theta/vr\cos(\psi-\psi_{p})\\ d\theta/ds=-L\cos\beta r/\big{(}v^{2}\cos\theta\cos(\psi-\psi_{p})\big{)}-1/ \cos(\psi-\psi_{p})+1/\big{(}v^{2}r\cos(\psi-\psi_{p})\big{)}+\Big{(}C_{\theta }+\widetilde{C}_{\theta}\Big{)}\\ d\psi/ds=-L\sin\beta r/\big{(}v^{2}\cos^{2}\theta\cos(\psi-\psi_{p})\big{)}- \sin\psi\tan\varphi/\big{(}\cos(\psi-\psi_{p})\big{)}+\Big{(}C_{\psi}+ \widetilde{C}_{\psi}\Big{)}\end{array}\right. \tag{5}\]  
The corresponding acceleration term can be converted into  
\[\left\{\begin{array}{l}C_{\theta}=-2r\omega_{e}\sin\psi\cos\varphi/\big{(}v \cos\theta\cos(\psi-\psi_{p})\big{)}\\ C_{\theta}^{\prime}=-\omega_{e}^{2}r^{2}(\cos\varphi\sin\varphi\cos\psi\sin \theta+\cos^{2}\phi\cos\theta)/\big{(}v^{2}\cos\theta\cos(\psi-\psi_{p})\big{)} \\ C_{\psi}=-2r\omega_{e}(\sin\varphi-\cos\varphi\tan\theta\cos\psi)/\big{(}v \cos\theta\cos(\psi-\psi_{p})\big{)}\\ C_{\psi}^{\prime}=-\omega_{e}^{2}r^{2}\sin\varphi\cos\varphi\sin\psi/v^{2}\cos ^{2}\theta\cos(\psi-\psi_{p})\end{array}\right. \tag{6}\]  
where \(\omega_{e}\) represents the earth rotation constant.