Chapter 7 - Extremal Trajectories in a Uniform Gravity Field

Abstract

This chapter presents the results of the studies of the minimum-fuel optimal control problem for a spacecraft moving in a uniform and drag-free gravity field with gravitational and thrust accelerations. The spacecraft is considered as a point mass with variable mass moving from a given initial manifold in the state space to a given landing point. Although the general theory of optimal trajectories is complete theoretically in the case of motion in the uniform gravity field, the number of studies on optimal analytical trajectory control solutions is very limited [3]. In this chapter, the analytical solutions for three-dimensional, extremal and optimal powered descent and landing trajectories are presented. It is shown that the optimality conditions and the analysis of canonical equations reveal five different optimal control regimes and corresponding behavior of the switching function and the cost function for each control regime. These regimes determine the control sequence and consequently, the 14-th-order canonical system of equations are integrated completely analytically in terms of time, thereby providing 14 new arbitrary integration constants.