  In this paper, we employ successive convexification to solve the minimum-time
6-DoF rocket powered landing problem. The contribution of this paper is the
development and demonstration of a free-final-time problem formulation that can
be solved iteratively using a successive convexification framework. This paper
is an extension of our previous work on the 3-DoF free-final-time and the 6-DoF
fixed-final-time minimum-fuel problems. Herein, the vehicle is modeled as a
6-DoF rigid-body controlled by a single gimbaled rocket engine. The trajectory
is subject to a variety of convex and non-convex state- and
control-constraints, and aerodynamic effects are assumed negligible. The
objective of the problem is to determine the optimal thrust commands that will
minimize the time-of-flight while satisfying the aforementioned constraints.
Solving this problem quickly and reliably is challenging because (a) it is
nonlinear and non-convex, (b) the validity of the solution is heavily dependent
on the accuracy of the discretization scheme, and (c) it can be difficult to
select a suitable reference trajectory to initialize an iterative solution
process. To deal with these issues, our algorithm (a) uses successive
convexification to eliminate non-convexities, (b) computes the discrete
linear-time-variant system matrices to ensure that the converged solution
perfectly satisfies the original nonlinear dynamics, and (c) can be initialized
with a simple, dynamically inconsistent reference trajectory. Using the
proposed convex formulation and successive convexification framework, we are
able to convert the original non-convex problem into a sequence of convex
second-order cone programming (SOCP) sub-problems. Through the use of Interior
Point Method (IPM) solvers, this sequence can be solved quickly and reliably,
thus enabling higher fidelity real-time guidance for rocket powered landings on
Mars.
