# Robust Near-Optimal Aerocapture Guidance Method Based on Saturation Function
## 3 Guidance Law Design Based on Saturation Function  
### Longitudinal Guidance Law  
The main goal of the longitudinal guidance law of aerocapture is to determine the magnitude of the bank angle to minimize the orbital insertion's velocity increment \(\Delta V\) after the atmospheric flight. To reduce the algorithm's complexity and make it possible to be applied online, the main control variable of the trajectory to be designed is the bank angle \(\sigma\), while the angle of attack \(\alpha\) follows a predetermined profile considering trim flight, thermal protection system, and so on [24]. According to [PERSON] [24], when lateral control is ignored in aerocapture (i.e., there is no constraint on the final orbit inclination), the two-phase bank angle control with bang-bang structure can achieve the minimum \(\Delta V\),  
\[|\sigma(t)|= \begin{cases}\sigma_{0},&0\leq t\leq t_{s}\\ \sigma_{d},&t>t_{s}\end{cases}  \tag{19}\]  
where \(t\) is the current time; \(\sigma_{0}\) is a selected small bank angle magnitude used in the first phase; \(\sigma_{d}\) is a relatively large bank angle magnitude used in the second phase; \(t_{s}\) denotes the switching time from \(\sigma_{0}\) to \(\sigma_{d}\). However, the guidance performance of this control profile is very sensitive to the selection of the switching time. To reduce the sensitivity to the corresponding parameter without losing too much optimality, a guidance profile based on the saturation function is proposed for predictor-corrector guidance. Firstly, the saturation function is introduced,  
\[s(t)=\tanh[k_{s}(t-t_{mid})] \tag{20}\]  
where \(t_{mid}\) denotes the time when the corresponding saturation function \(s(t)\) reaches the median, and it's the guidance parameter that must be determined; \(k_{s}\) can be preset as a constant value. Considering that the value range of \(s(t)\) is \([-1,1]\), let \(\sigma_{\text{max}}\) and \(\sigma_{\text{min}}\) denote the upper and lower bounds of bank angle magnitude, respectively, then the control profile based on \(s(t)\) can be written as  
\[ \begin{split}|\sigma_{\text{s}}(t)|&=\frac{\sigma_ {\text{max}}+\sigma_{\text{min}}}{2}+\frac{\sigma_{\text{max}}-\sigma_{\text{ min}}}{2}s(t)\\ &=\frac{\sigma_{\text{max}}-\sigma_{\text{min}}}{2}\left\{\frac{ \sigma_{\text{max}}+\sigma_{\text{min}}}{\sigma_{\text{max}}-\sigma_{\text{ min}}}+s(t)\right\}\end{split}  \tag{21}\]  
Figure 2: Two-impulse aerocapture maneuver: (**a**) \(r_{a0}<r_{alg}t\); (**b**) \(r_{a0}>r_{alg}\). At the same time, a linear function is introduced to prevent premature saturation of bank control so that the guidance law can have more control margin to cope with uncertainties and disturbances,  
\[l(t)=y_{l}+k_{l}(t-t_{mid}) \tag{22}\]  
where \(k_{l},y_{l}\) can be preset as constant values. The final specific guidance profile is  
\[ \begin{split}|\sigma(t)|&=|\sigma_{\text{s}}(t)|l(t) \\ &=\frac{\sigma_{\text{max}}-\sigma_{\text{min}}}{2}\bigg{\{}\frac{ \sigma_{\text{max}}+\sigma_{\text{min}}}{\sigma_{\text{max}}-\sigma_{\text{min} }}+s(t)\bigg{\}}l(t)\\ &=\frac{\sigma_{\text{max}}-\sigma_{\text{min}}}{2}\bigg{\{}\frac {\sigma_{\text{max}}+\sigma_{\text{min}}}{\sigma_{\text{max}}-\sigma_{\text{min }}}+\tanh[k_{s}(t-t_{\text{mid }})]\bigg{\}}[y_{l}+k_{l}(t-t_{\text{mid }})]\end{split}  \tag{23}\]  
The proposed guidance profile has mainly three advantages. First, it is similar to the optimal bang-bang control due to the introduction of saturation function, so the guidance method based on this profile can realize near-optimal performance. Second, different from the instantaneous and discontinuous switching in the bang-bang profile, the control magnitude's switching in this profile based on saturation function is smoother and easier to adjust, which is conducive to reducing the aforementioned parameter sensitivity and improving the guidance method's robustness.