Estimação da trajetória ótima de um lançador movido a foguete de propelente sólido

Abstract

In this work the optimal trajectory of orbital insertion of a Rocket-Powered Launcher is calculated, considering the discontinuities caused by the stage shifts, via the optimal control theory. First the translational dynamics of each stage is obtained. Then, the optimal control theory is developed, for problems with discontinuities and defined by various dynamics, using the calculus of variations. This theory transforms the problem of finding a control function u(t), which minimizes a performance index, in a Multipoint Boundary Value Problem. For nonlinear dynamics, as is the case, the solution of this problem can usually only be obtained through numerical methods. Due to the nature of the problem, two numerical methods had to be used, the Collocation Method, already implemented in the MATLAB platform, and the Multiple Shooting Method, the first is used due to its larger converge region and the second due to its ease of parallelization, which considerably reduces the time needed to a solution. The problem is then solved with the aid of the Continuation Method, which consists of gradually deforming the solution of a known simpler problem into the final and more complex problem desired. All the methods and algorithms used are presented and finally the optimal trajectory found is presented and discussed.