• first stage
• mathematical model
• re-entry dynamics
• reusable space launcher
• state-space model

The development of reusable launchers has increased significantly in recent years, mainly due to the advent of private companies such as SpaceX and Blue Origin that have achieved great successes in this field. The main reasons for going down this path are to reduce the cost and environmental impact of the space industry.
This paper addresses the study of the re-entry dynamics and control of the first stage of a reusable launcher schematized as a rigid body having 3 DOF, two translations in the vertical plane and the rotation in pitching.
In the first part, the dynamic is faced by solving nonlinear coupled differential equations under the assumption of no active control by engine throttle, engine gimbal angle, fins deflection and cold gas thrusters activation. The equations are written with respect to an inertial reference system centered in the Earth, which is assumed to be non-rotating. The study begins from the flip maneuver, that takes place above the Kàrmàn line, and finishes with the stage landing. Special attention is given to critical issues that may arise during the last kilometers of the atmospheric re-entry and in particular to the behavior of the stage attitude angle.
In the second part, stability in dense atmosphere is investigated through a model in state space. Specifically, system responses at various altitudes to the movement of the fins, to the activation of the lateral thrusters and to a step wind gust disturbance.
Among the most interesting results are the time history of velocity, attitude and altitude, achieved by simulations in MATLAB environment, as well as the mathematical explicitation of the transfer functions of the system in state space for the last tracts of the mission. The latter result is the most important as it makes the model general and allows the study of a number of cases of interest as the inputs to the system change.