## 3 Convexification and Discretization of the Re-Entry Trajectory Generation Problem
### State, Control Variable Settings and Model Convexity  
According to the equation dynamic model (5), the control variables are the attack angle and bank angle, while the lift and drag coefficients of the CAV [29], like the vehicle simulated in this paper generally meet the following equations:  
\[\left\{ \begin{array}{l}C_{L}=c_{l2}\alpha+c_{l1}\\ C_{D}=c_{d3}\alpha^{2}+c_{d2}\alpha+c_{d1}\end{array} \right. \tag{13}\]  
where, \(c_{l1}\), \(c_{l2}\) represent the lift parameters, and \(c_{d1}\), \(c_{d2}\), \(c_{d3}\) represent the drag parameters. Set the state variable \(X=[r,\phi,\varphi,v,\theta,\psi,\alpha,\beta]\) and control variable \(U=[\dot{\alpha},\dot{\beta}]\) of the vehicle. With the new control quantity, the dynamic model (5) is converted into  
\[\left\{\begin{array}{l}dr/ds=-r\tan\theta/\cos(\psi-\psi_{p})\\ d\phi/ds=-\sin\psi/\left(\cos\varphi\cos(\psi-\psi_{p})\right)\\ d\varphi/ds=-\cos\psi/\cos(\psi-\psi_{p})\\ d\psi/ds=C_{D}rqs/\left(mg_{0}v^{2}\cos\theta\cos(\psi-\psi_{p})+\tan\theta/ \left(vr\cos(\psi-\psi_{p})\right)\right.\\ \left. \begin{array}{l}-1/\cos(\psi-\psi_{p})+1/\left(v^{2}r\cos(\psi-\psi_{ p})\right)+\left(C_{\theta}+\widetilde{C}_{\theta}\right)\\ d\psi/ds=-C_{L}\sin\beta rqs/\left(mg_{0}v^{2}\cos\theta\cos(\psi-\psi_{p}) \right)\\ \left.\begin{array}{l}-\sin\psi\tan\varphi/\left(\cos(\psi-\psi_{p})\right)+ \left(C_{\psi}+\widetilde{C}_{\psi}\right)\\ d\alpha/ds=\dot{\alpha}\\ d\beta/ds=\dot{\beta}\end{array}\right.\end{array} \right. \tag{14}\]  
Equation (14) can be expressed as  
\[\dot{X}=g(X,S)+BU(S)+h(X,S)=F(X,U,S) \tag{15}\]\[g(X,S)=\left[ \begin{array}{l}-r\tan\theta/\cos(\psi-\psi_{p})\\ -\sin\psi/\left(\cos\varphi\cos(\psi-\psi_{p})\right)\\ -\cos\psi/\cos(\psi-\psi_{p})\\ C_{D}\tau qs/mg_{0}v\cos\theta\cos(\psi-\psi_{p})+\tan\theta/\left(vr\cos( \psi-\psi_{p})\right)\\ -C_{L}\cos\beta rqs/mg_{0}v^{2}\cos\theta\cos(\psi-\psi_{p})-\frac{r}{v^{2} \cos(\psi-\psi_{p})}(\frac{v^{2}}{r}-\frac{1}{r^{2}})\\ -C_{L}\sin\beta rqs/mg_{0}v^{2}\cos^{2}\theta\cos(\psi-\psi_{p})-\sin\psi\tan \varphi/\cos(\psi-\psi_{p})\\ 0\\ 0\\ \end{array} \right] \tag{16}\]  
\[B=\left[ \begin{array}{cccccccc}0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1\end{array} \right]^{T} \tag{17}\]  
\[h(X,S)=\left[0\;0\;0\;0\;C_{\theta}+C_{\theta}^{\prime}\;C_{\psi}+C_{\psi}^{ \prime}\;0\;0\;\right]^{T} \tag{18}\]  
Since \(g(X,S)\) is a nonlinear function of \(X\), further linearization of the dynamic equation is required. The first order Taylor expansion is operated on the last iterative trajectory, which can be expressed as  
\[\dot{X}=\left.\frac{\partial g(X)}{\partial X}\right|_{x=x^{(k)}}X+BU+g\Big{(} X^{(k)}\Big{)}-\left.\frac{\partial g(X)}{\partial X}\right|_{X=X^{(k)}}X^{(k)}+h(X ^{(k)}) \tag{19}\]  
Let \(A=\left.\frac{\partial g(x)}{\partial x}\right|_{x=x^{(k)}}\), \(C=g\Big{(}X^{(k)},S\Big{)}-AX^{(k)}+h(X^{(k)},S)\), then  
\[\dot{X}=AX+BU+C \tag{20}\]  
\[A=\left[ \begin{array}{cccccccc}a_{11}&0&0&0&a_{15}&a_{16}&0&0\\ 0&0&a_{23}&0&0&a_{26}&0&0\\ 0&0&0&0&0&a_{36}&0&0\\ a_{41}&0&0&a_{44}&a_{45}&a_{46}&a_{47}&0\\ a_{51}&0&0&a_{54}&a_{55}&a_{56}&a_{57}&a_{58}\\ a_{61}&0&a_{63}&0&a_{65}&a_{66}&a_{67}&a_{68}\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{array} \right] \tag{21}\]  
\[\left\{ \begin{array}{l}a_{11}=-\tan\theta^{(k)}/\cos\Bigl{(}\psi^{(k)}-\psi _{p}\Bigr{)}\\ a_{41}=D^{(k)}\Bigl{(}1-\mathrm{R}_{e}r^{(k)}/H_{0}\Bigr{)}/\Bigl{(}v^{(k)} \cos\theta^{(k)}\cos\Bigl{(}\psi^{(k)}-\psi_{p}\Bigr{)}\Bigr{)}\\ -\tan\theta^{(k)}/v^{(k)}r^{(k)2}\cos\Bigl{(}\psi^{(k)}-\psi_{p}\Bigr{)}\\ a_{51}=-L^{(k)}\cos\beta^{(k)}\Bigl{(}1-\mathrm{R}_{e}r^{(k)}/H_{0}\Bigr{)}/ \Bigl{(}v^{2(k)}\cos\theta^{(k)}\cos\Bigl{(}\psi^{(k)}-\psi_{p}\Bigr{)}\Bigr{)} \\ -1/v^{(k)2}r^{(k)2}\cos\Bigl{(}\psi^{(k)}-\psi_{p}\Bigr{)}\\ a_{61}=-L^{(k)}\sin\beta^{(k)}\Bigl{(}1-\mathrm{R}_{e}r^{(k)}/H_{0}\Bigr{)}/ \Bigl{(}v^{2(k)}\cos\theta^{2(k)}\cos\Bigl{(}\psi^{(k)}-\psi_{p}\Bigr{)}\Bigr{)} \\ \end{array} \right. \tag{22}\]  
\[\left\{