# Robust Near-Optimal Aerocapture Guidance Method Based on Saturation Function
## 3 Guidance Law Design Based on Saturation Function
### Lateral Guidance Law  
Unlike the longitudinal motion, which determines the orbital shape, the lateral motion mainly determines the direction of orbital angular momentum, which is reflected in the orbital inclination \(i\). This paper adopts a predictive single bank reversal logic: in each guidance cycle, we assume that the bank angle is reversed at the current time, then run the predictive guidance scheme (as mentioned above), and obtain the post-aerocapture orbit inclination \(i_{pre}\). If \(i_{pre}\) is equal to the expected orbit inclination \(i_{tgt}\) or meets the accuracy requirements,  
\[\big{|}i_{pre}-i_{tgt}\big{|}\leq\varepsilon \tag{24}\]  
then we reverse the bank angle. It should be noted that the bank angle rate and acceleration limits are applied to the numerical integration process and ensure that the spacecraft rolls along the shortest path to reduce the impact on longitudinal motion.  
Moreover, as a model-based algorithm, the uncertainty of aerodynamic coefficients (\(C_{L}\) and \(C_{D}\)), atmospheric density, and vehicle mass will significantly affect the performance of the aerocapture guidance algorithm [32]. If the estimated information of aerodynamic coefficients and atmospheric density deviation can be used reasonably, the algorithm will achieve the desired high guidance accuracy, especially when the aerocapture process is under severe uncertainties and disturbances. Therefore, it is necessary for the guidance system to make appropriate online corrections according to the aerodynamic parameters measured in real time.  
Aerocapture's dispersions include the dispersions of entry states, aerodynamics, spacecraft mass, and atmospheric density. Except for the dispersions of entry states, all dispersion only affects the spacecraft's lift and drag accelerations. During flight, the actual values of lift and drag accelerations can be calculated from navigation data and accelerometer output. In the absence of other measurement means (such as the accurate measurement system of atmospheric data), it is impossible to distinguish the influence of a dispersion source (such as density) from others (such as aerodynamic coefficients). However, it needs only to estimate the comprehensive effect of these dispersions. Therefore, a simple first-order low-pass filter can be used to estimate the dispersion effect and smooth the measured data:  
\[X_{n+1}=X_{n}+(1-K_{g})(X^{*}-X_{n}) \tag{25}\]  
where \(X^{*}\) is the ratio of the current measured value (lift or drag acceleration) to the corresponding nominal value obtained from the nominal model, \(X_{n}\) is the filter ratio of the last guidance cycle, and \(K_{g}\) is a gain in the range (0, 1). The larger the gain, the better the filter's smoothness, but the lower the sensitivity; Otherwise, the sensitivity is higher, but the stability is worse. Generally, a value slightly less than 1 is taken so that the impact of the previous round of filtration ratio \(X_{n}\) on the current filtration ratio \(X_{n+1}\) is greater. In this paper, \(K_{g}=0.9\), and the sampling time of the filter can be equal to the guidance period.  
When initializing the filter, \(X_{0}\) is set to 1. In each guidance cycle, the filter's output is multiplied by the nominal lift \(L\) (or drag \(D\)) profile, and the subsequent trajectory is calculated based on the corrected profile.