# RCS and Aero Surfaces Control Allocation Research on RLV's Re-Entry Phase
## 2 Mathematical Model
### Dynamic Model of RLV  
The 6-DOF dynamic equations of the rigid-body RLV are made up of the 3-DOF translational equations and the 3-DOF rotational equations. The 3-DOF translational equations in the launch inertial coordinate system, during the re-entry phase, are given as follows  
\[m\left[\begin{array}{c}\dot{V}\\ V\dot{\theta}\cos\psi_{v}\\ -V\dot{\psi}_{v}\end{array}\right]=\mathbf{C_{n}^{hw}}\left[\begin{array}{c}0 \\ -mg\\ 0\end{array}\right]+\mathbf{C_{v}^{hw}R}+\mathbf{F_{RCS}} \tag{3}\]  
where \(m\) is the mass of RLV; \(\theta\) is the flight-path angle; \(\psi_{v}\) is the flight deflection angle; \(\gamma_{v}\) is the velocity bank angle; \(\mathbf{C_{n}^{hw}}\) is the transformation matrix from the launch inertial to the ballistic coordinate system; and \(\mathbf{C_{v}^{hw}}\) is the transformation matrix from the velocity coordinate system to the ballistic coordinate system. The specific expansion of \(\mathbf{C_{n}^{hw}}\), \(\mathbf{C_{v}^{hw}}\) are described as  
\[\mathbf{C_{n}^{hw}}=\left[\begin{array}{ccc}\cos\theta\cos\psi_{v}&\sin \theta\cos\psi_{v}&-\sin\psi_{v}\\ -\sin\theta&\cos\theta&0\\ \cos\theta\sin\psi_{v}&\sin\theta\sin\psi_{v}&\cos\psi_{v}\end{array}\right], \mathbf{C_{v}^{hw}}=\left[\begin{array}{ccc}1&0&0\\ 0&\cos\gamma_{v}&-\sin\gamma_{v}\\ 0&\sin\gamma_{v}&\cos\gamma_{v}\end{array}\right] \tag{4}\]  
The 3-DOF rotational equations used for [PERSON]'s attitude control, during the re-entry phase, are built in the body coordinate system, and expressed as  
\[\left[\begin{array}{c}\dot{\gamma}\\ \dot{\psi}\\ \dot{\varphi}\end{array}\right]=\left[\begin{array}{ccc}1&\sin\gamma\tan \psi&\cos\gamma\tan\psi\\ 0&\cos\gamma&-\sin\gamma\\ 0&\sin\gamma/\cos\psi&\cos\gamma/\cos\psi\end{array}\right]\left[\begin{array} []{c}\omega_{x}\\ \omega_{y}\\ \omega_{z}\end{array}\right],\left[\begin{array}{c}\dot{\omega}_{x}\\ \dot{\omega}_{y}\\ \dot{\omega}_{z}\end{array}\right]=-\mathbf{I}_{0}^{-1}\mathbf{\Omega}_{0}^{- 1}\left[\begin{array}{c}\omega_{x}\\ \omega_{y}\\ \omega_{z}\end{array}\right]+\mathbf{I}_{0}^{-1}(\mathbf{M}_{R}+\mathbf{M}_{ RCS}) \tag{5}\]  
where \(\gamma,\psi,\varphi\) are the roll, yaw, and pitch angle, respectively; \(\omega_{x},\omega_{y},\omega_{z}\) are the roll, yaw, and pitch angular velocity of RLV's body coordinate rotation from the launch inertial coordinate, respectively; \(\mathbf{I}_{0}\) are the moment matrix of inertia. The expansions of \(\mathbf{\Omega}\) is  
\[\mathbf{\Omega}=\left[\begin{array}{ccc}0&-\omega_{z}&\omega_{y}\\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\end{array}\right] \tag{6}\]  
According to the aforementioned equations, the dynamic model of RLV can be written as  
\[\dot{\mathbf{x}}\mathbf{=}\mathbf{A}\mathbf{x+}\mathbf{B}\mathbf{u} \tag{7}\]  
where the state vector is defined as \(\mathbf{x}=\left[\gamma,\psi,\alpha,\omega_{x},\omega_{y},\omega_{2}\right]^ {T}\); the control input vector is defined as \(\mathbf{u}=\left[\underbrace{v_{1},v_{2},\cdots,v_{N}}_{RCS},\underbrace{ \delta_{da},\delta_{dr},\delta_{de},\delta_{df}}_{AS}\right]\); the system matrix \(\mathbf{A},\mathbf{B}\) can be derived by Equations (3) and (5).