# Rapid Construction of Aerocapture Attainability Sets Using Sequential Convex Programming
## 5 Numerical Demonstrations
### Influence of Path Constraints to the Aerocapture Attainability Sets  
Heating rate, load factor, and dynamic pressure [53]  
\[\dot{Q}=9.4369\times 10^{-9}\Big{(}\sqrt{g_{M}R_{M}}\Big{)}^{3.15}\sqrt{\rho}V^{3. 15}\leq\dot{Q}_{\max} \tag{31}\]  
\[n=\sqrt{L^{2}+D^{2}}\leq n_{\max} \tag{32}\]  
\[q=0.5\times 10^{-3}g_{M}R_{M}\rho V^{2}\leq q_{\max} \tag{33}\]  
are important path constraints during the atmospheric flight. The heating rate \(\dot{Q}\) of the stagnation point on the surface of the vehicle is in watts per square centimeter. The load factor \(n\) is in \(g_{M}\). The dynamic pressure \(q\) is in kilonewtons per square meter. The unit of dimensional atmospheric density \(\rho\) is in kilograms per cubic meter, and the velocity \(V\) is nondimensional. Correspondingly, \(\dot{Q}_{\max}\), \(n_{\max}\), and \(q_{\max}\) are the upper limits of the heating rate, load factor, and dynamic pressure limits, respectively. Given that \(V=\sqrt{2/r^{2}-2e}\), the constraints are rewritten in the form of radial distance as  
\[r(e)\geq r_{\dot{Q}}(e)=\frac{-2h_{s}}{R_{M}}\ln\frac{\dot{Q}_{\max}}{9.4369 \times 10^{-9}(\sqrt{g_{M}R_{M}})^{3.15}\sqrt{\rho_{0}V^{3.15}}}+1 \tag{34}\]  
\[r(e)\geq r_{n}(e)=-\frac{h_{s}}{R_{M}}\ln\frac{2 mnt_{\max}}{R_{M}\sqrt{C_{L}{}^ {2}+C_{D}{}^{2}}\rho_{0}V^{2}}+1 \tag{35}\]  
\[r(e)\geq r_{q}(e)=-\frac{h_{s}}{R_{M}}\ln\frac{2\times 10^{3}q_{\max}}{g_{M}R_{M }\rho_{0}V^{2}}+1 \tag{36}\]  
where \(r_{\dot{Q}}(e)\), \(r_{n}(e)\), and \(r_{q}(e)\) are the minimum allowed radial distances for a given e corresponding to the upper limits of heating rate, load factor, and dynamic pressure, respectively. These path constraints are linear inequality constraints and must be all satisfied in the solving of _Problem 1_ with convex optimization. The path constraints in the simulation were \(\dot{Q}_{\max}\) = 105 W/cm\({}^{2}\), \(n_{\max}\) = 4.5 \(g_{M}\), and \(q_{\max}\) = 10 KN/m\({}^{2}\). Note that the upper boundary of the path constraint usually depends on the corresponding endurance of the spacecraft. For a low lift-to-drag ratio spacecraft, the heating rate constraint is generally between 30 and 300 W/cm\({}^{2}\), and the maximum overload peak value does not exceed 20 \(g_{M}\). The path constraint boundary values used in this paper refer to the design values of most existing Mars rovers. The resulting aerocapture attainability sets with and without path constraints are plotted in Figure 13. It shows that path constraints significantly reduce the range of the aerocapture attainability set. The peak load factor, heating rate, and dynamic pressure together with the atmospheric flight time of aerocapture for each of the points \(A\), \(B\), \(C\) and \(C^{\prime}\) are given in Table 2. The bank angle profiles of the four characteristic points are shown in Figure 14. Despite the case of \(D\leq\) 0.1 \(g_{M}\), wherein the states are integrated directly with a zero bank angle, the bank angle in the optimal profile switches twice, and the \(\sigma_{\max}\) always appears in the middle arc. In point \(C^{\prime}\), a singular arc of the bank angle profile exists, that is, the bank angle profile is not exactly the bang-bang control structure used to satisfy the path constraints. The comparison of the path variable profiles between points \(C\) and \(C^{\prime}\) is shown in Figure 15, which intuitively reflects the effect of suppressing the path variables. Generally, large peak values of the path variables may threaten the aerocapture vehicle; however, the results in this subsection indicate that the peak values of the path variables are tolerable for Mars aerocapture. Figure 14: Bank angle profiles for each of the points \(A\), \(B\), \(C\), and \(C^{\prime}\) in \[n=\sqrt{L^{2}+D^{2}}\leq n_{\max} \tag{32}\]0