# Rapid Construction of Aerocapture Attainability Sets Using Sequential Convex Programming
## 1 Introduction  
For target planets with an atmosphere, such as Venus, Mars, Titan, and Neptune [1, 2, 3], aerocapture is an orbital maneuver that captures an interplanetary probe from a hyperbolic orbit or interplanetary approach orbit and places it in a mission orbit around the target planet through planetary atmospheric flight. Compared with traditional propulsive capture, aerocapture can significantly reduce the required fuel consumption [4], which attracts a lot of research attention [5, 6, 7, 8]. The high deceleration efficiency of aerocapture maneuver renders a wide range of potential applications. Typical applications are shown schematically in Figure 1. An aerocapture maneuver can capture a vehicle in a planetary orbit or deliver the vehicle into the periodic orbits around the Sun-planet libration points, such as collinear libration points 1 and 2 (L1 and L2) [9, 10, 11, 12, 13, 14, 15, 16]. In addition, an aerocapture maneuver can adjust the phase and flight velocity of an interplanetary probe to fly along an ideal trajectory. Such an aerocapture mode is called aerogravity assist (AGA) [17, 18, 19]. Note that in both cases, the target orbit is eventually reached after aerocapture through an intermediate transfer orbit, such as the stable manifold of a halo orbit or an interplanetary transfer orbit. These transfer orbits are determined directly by the terminal state of the atmospheric flight. Therefore, accurately constructing the aerocapture attainability sets is crucial in assessing the vehicle's reachable ability after aerocapture. An aerocapture attainability set describes the maneuverability of the aerocapture and determines the feasibility of a given aerocapture mission. Constructing the aerocapture attainability set is equivalent to determining the boundary of the attainability set, and numerically solving the aerocapture trajectory for each terminal state on the boundary is a family of optimal control problems. Therefore, a robust and efficient algorithm of aerocapture trajectory planning is key to accurately construct and formulate the complete aerocapture attainability set. To solve the aerocapture trajectory planning problem, many guidance algorithms have been proposed since the 1980s, such as the Apollo entry guidance scheme [20], analytic aero capture guidance [21; 22; 23; 24], and numerical predictive-corrector aerocapture guidance [25; 26; 27; 28; 29]. They are reliable but not globally optimal. At the same time, open-loop aerocapture trajectory optimization methods were also investigated [30; 31; 32; 33]. In comparison with the guidance algorithms, optimization methods ensure the optimal characteristic of aerocapture trajectories. The aerocapture trajectory can be optimized by an indirect or direct method, such as the direct collocation pseudospectral method [34]. However, the open-loop optimization is slow in computation speed and has a worse convergence [35; 36]. Consequently, existing aerocapture trajectory planning methods, both the guidance algorithms and the open-loop optimization methods, are not suitable for the rapid and accurate construction of an aerocapture attainability set. This paper develops a robust and efficient algorithm based on convex optimization. Convex optimization is a popular method for rapid planning and has been used widely in aerospace in recent years. Typical applications include planetary powered descent guidance [37; 38], space rendezvous and proximity operations [39], planetary entry trajectory planning [40], and path planning for zero-propellant maneuvers [41; 42]. If a nonlinear programming problem can be formulated to a specific form of convex optimization, such as linear programming, second-order cone programming, and semidefinite programming [43], the optimal solution of the nonlinear problem can be found rapidly and accurately [44]. In Section 2, the aero capture attainability set is defined in the longitudinal dynamics model, and the problem is formulated as determining the boundary of the attainability set.