2023-07-24 22:00:00
The problem of finding the optimal thrust level of a rocket engine has been studied intensively since the seminal work of Goddard in the early 20th century. The classical formulation assumes that the level of thrust varies freely between zero and a given maximum value. The optimal solution is then composed of maximum thrust bows and ballistic bows.
This article presents an alternative hypothesis which consists of a parametric thrust profile, either linear or with two steps. This modeling is representative of the operation of most engines. The objective is to simultaneously optimize the thrust profile and the trajectory of the last stage to reach the targeted orbit while minimizing propellant consumption. The initial conditions are given and result from the flight of the lower stages. The target orbit is an ellipse of given apogee and perigee. It is assumed that the trajectory of the last stage is planar, which is the case in the majority of practical applications.
This hybrid optimal control problem including the thrust profile is of great practical importance when designing new engines. The study of the optimality conditions shows that this problem admits an analytical solution in state feedback, the optimal thrust direction being expressed as a function of the current kinematic conditions (position, speed). This study also shows that the optimal injection point is the perigee of the targeted orbit and that the orbit will be reached by a descending phase. This theoretical result explains the a priori surprising shape of many launch trajectories, first rising, then descending. It turns out that this form is in fact associated with a choice of thrust level close to the optimum. These properties make it possible to reduce the problem of optimal control to a nonlinear problem with two variables, whose numerical solution is simple.
The theoretical study also provides an analytical expression of the adjoint vector as a function of position and velocity conditions. Adjoint vector initialization is a hard point for optimal spatial trajectory problems. It turns out that this analytical initialization, obtained when the thrust level is optimized, is effective in solving different problems, in particular when the engine thrust level is fixed.
This article presents the formulation of the optimal control problem and the theoretical study of the optimality conditions. The different properties of the optimal trajectory are established, then used to solve practical application cases.
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#Space #trajectories #Injection #orbit #optimum #thrust #level #Complete #file
