# Optimal Gliding Trajectories for Descent on Mars
## 1 Introduction  
The terminal phase of an interplanetary mission to deploy a payload on the surface of Mars is composed of the entry, descent, and landing (EDL). This is a critical phase of the mission due to the spacecraft-atmosphere interaction, which increases aerodynamic and heat loads on the structure of the spacecraft due to deceleration from high hypersonic velocities, mainly dominated by drag. When compared to Earth's atmosphere, the heat transfer and loads are lower on Mars due to its composition and the lower mass of the planet, which reduces the gravitational attraction at the descent. Spacecraft-like spaceplanes have been useful for re-entry on Earth, saving energy because they do not require the action of the propulsion system to glide, such as thrusters. At the same time, the spaceplanes are qualified for space activities; two examples of these are the space shuttles and the robotic X-37. Their large area-to-mass ratio (\(A/m\)), high aerodynamic performance, and the implementation of aerodynamic control surfaces allow gliding flight above the planet, increasing the range and cross-range of the trajectory. This property facilitates the direction of the spaceplane to a specific point for the final approach and landing, improving the controllability of the trajectory, different from the descent capsules, which do not have surfaces to increase the endurance of the atmospheric flight. The optimization of the trajectory focuses on the satisfaction of a specific cost function along or at the end of the flight. A way to calculate optimal trajectories is through an optimal control problem (OCP). This method is highly implemented in the aerospace industry, robotics, and autonomous vehicles to compute the inputs of manipulated variables to control the state vector of the resulting trajectory to minimize or maximize the cost function. In the case of gliding spaceplanes, the idea is to find the history of the aerodynamic angles (angle of attack and banking angle) to direct the spacecraft onto the optimal path. Various research has presented the benefits of optimal trajectories for interplanetary maneuvers passing over the atmosphere of a celestial body [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. A way of optimizing the aeromaneuver of a hypersonic glider is via Pontryagin's maximum, which is used to minimize the loss in energy, increasing the turn angle or atmospheric flight around Venus and Mars [4]. This is an example of optimization at a terminal time. Another kind of optimization, along the trajectory, was explored with the addition of a heat constraint, which reduces the velocity because it is proportional to the increase in heat [5; 11]. When the trajectory is analyzed outside of the plane, control of the bank angle is required. This is advantageous in changing the inclination of the heliocentric orbit [6]. Other constraints were explored in [7], dealing with a traditional optimal control problem, different from [8], where multi-objective genetic algorithms were implemented, looking for a minimization of the Delta-V given by the propellant. To solve this problem, nonlinear programming (NLP) in an OCP was applied to optimal gravity-assist maneuver above Mars [1]. Optimal guidance was studied in [9]. The cost function to maximize the turn angle was presented in [10], and the optimization of a powered gravity assist was analyzed in [12]. Optimal trajectories above Venus, Earth, and Mars were studied to maximize the terminal velocity, latitude, and longitude, controlling the aerodynamic angles of attack and baking on a spaceplane at high altitudes at the beginning of the continuum flow [13; 14]. Optimal gliding descent of hypersonic vehicles has been analyzed in some scientific papers; for instance, a second-order cone programming to optimize entry trajectories of hypersonic gliders with high lift/drag ratios was explored in [15].