# Robust Near-Optimal Aerocapture Guidance Method Based on Saturation Function
## 2 Aerocapture Modeling and Maneuver Description  
### Three-DOF Dynamic Model  
During aerocapture, without loss of generality, we can assume that the celestial body is an oblate sphere rotating with constant angular velocity \(\omega\), and its atmosphere is at rest with respect to the celestial body. Then the three-dimensional equations describing the motion of the spacecraft are  
\[\dot{r}=V\sin\gamma \tag{1}\]  
\[\dot{\theta}=\frac{V\cos\gamma\sin\psi}{r\cos\phi} \tag{2}\]  
\[\dot{\phi}=\frac{V\cos\gamma\cos\psi}{r} \tag{3}\]  
\[\dot{V}=-D-g_{r}\sin\gamma-g_{\phi}\cos\gamma\cos\psi \tag{4}\]  
\[\begin{split}&+\omega^{2}r\text{cos}\phi(\sin\gamma\text{ cos}\phi-\cos\gamma\cos\psi\sin\phi)\\ \dot{\gamma}=&\frac{1}{V}\Big{[}L\text{cos}\sigma+ \Big{(}\frac{V^{2}}{r}-g_{r}\Big{)}\text{cos}\gamma+g_{\phi}\sin\gamma\text{ cos}\psi\\ &+2\omega V\sin\psi\cos\phi+\omega^{2}r\text{cos}\phi(\cos \gamma\text{cos}\phi+\sin\gamma\text{cos}\psi\phi)]\\ \dot{\psi}=&\frac{1}{V}\bigg{[}\frac{L\text{sin} \sigma}{\cos\gamma}+\bigg{(}\frac{(V\text{cos}\gamma)^{2}\text{tan}\phi}{r}+g_ {\phi}\bigg{)}\frac{\sin\psi}{\cos\gamma}\\ &-2\omega V(\tan\gamma\cos\psi\text{cos}\phi-\text{sin}\phi)+ \frac{\omega^{2}r}{\cos\gamma}\sin\psi\sin\phi\text{cos}\phi\bigg{]}\end{split} \tag{6}\]  
where \(r\) is the radial distance from the center of the celestial body to the spacecraft's center of mass; \(\theta\) and \(\phi\) are longitude and latitude, respectively; \(V\) refers to the velocity of the spacecraft relative to the celestial body; \(\gamma\) is the flight path angle of the velocity vector relative to the celestial body; \(\psi\) denotes the heading angle (turning clockwise from the due north, the projection of the relative velocity vector of the celestial body on the local horizontal plane is positive); \(\sigma\) refers to the bank angle, \(g_{r}\) and \(g_{\phi}\) are the gravitational acceleration's radial and latitudinal components, respectively, which are calculated as follows,  
\[g_{r}=\frac{\mu}{r^{2}}\left[1+J_{2}\left(\frac{R_{0}}{r}\right)^{2}\left(1.5-4.5\mathrm{sin}^{2}\phi\right)\right] \tag{7}\]  
\[g_{\phi}=\frac{3\mu}{r^{2}}J_{2}\left(\frac{R_{0}}{r}\right)^{2}\sin\phi \mathrm{cos}\phi \tag{8}\]  
where \(\mu\) is the gravitational parameter of the celestial body, \(R_{0}\) is the equatorial radius of the celestial body, and \(J_{2}\) is the second zonal coefficient (for oblateness). In addition, the aerodynamic lift and drag accelerations (\(L\) and \(D\)) are conventionally defined as  
\[L=\frac{1}{2}\rho V^{2}\frac{S}{m}C_{L} \tag{9}\]  
\[D=\frac{1}{2}\rho V^{2}\frac{S}{m}C_{D} \tag{10}\]  
where \(\rho\) is the atmospheric density; \(S\) denotes the reference area of the spacecraft; \(C_{L}\), \(C_{D}\) are lift coefficient and drag coefficient respectively; \(m\) is the mass of the spacecraft. In this work, the Martian atmosphere uses the atmospheric exponential model fitted according to the inflight measurement data of Viking-1 launched by NASA in 1976 as the nominal density model,  
\[\rho=\rho_{0}e^{-h/h_{s}} \tag{11}\]  
where, the reference density \(\rho_{0}=1.474\times 10^{7}\) kg/km\({}^{3}\), the scale altitude \(h_{s}=8.8057\) km, \(h\) is the altitude of the spacecraft. While the Earth's atmosphere uses the 1976 U.S. standard atmosphere model as the nominal density model. The orbital maneuver above the celestial atmosphere is calculated by the spacecraft's inertial velocity expressed in the equatorial inertial coordinate system. The inertial velocity vector \(\mathbf{V}_{i}\) is equal to the sum of the relative velocity vector \(\mathbf{V}\) and the cross product of the celestial body's self-rotation angle velocity vector \(\mathbf{\omega}\) and the spacecraft's position vector \(\mathbf{r}\):  
\[\mathbf{V}_{i}=\mathbf{V}+\mathbf{\omega}\times\mathbf{r} \tag{12}\]