  In this paper we develop a geometric analysis and a numerical algorithm,
based on indirect methods, to solve optimal guidance of endo-atmospheric launch
vehicle systems under mixed control-state constraints. Two main difficulties
are addressed. First, we tackle the presence of Euler singularities by
introducing a representation of the configuration manifold in appropriate local
charts. In these local coordinates, not only the problem is free from Euler
singularities but also it can be recast as an optimal control problem with only
pure control constraints. The second issue concerns the initialization of the
shooting method. We introduce a strategy which combines indirect methods with
homotopies, thus providing high accuracy. We illustrate the efficiency of our
approach by numerical simulations on missile interception problems under
challenging scenarios.
