  In this paper we present a new approach to solve the fuel-efficient powered
descent guidance problem on large planetary bodies with no atmosphere (e.g. the
Moon or Mars) using the recently developed Theory of Functional Connections.
The problem is formulated using the indirect method which casts the optimal
guidance problem as a system of nonlinear two-point boundary value problems.
Using the Theory of Functional Connections, the problem constraints are
analytically embedded into a "constrained expression," which maintains a
free-function that is expanded using orthogonal polynomials with unknown
coefficients. The constraints are satisfied regardless of the values of the
unknown coefficients which convert the two-point boundary value problem into an
unconstrained optimization problem. This process casts the solution into the
admissible subspace of the problem and therefore simple numerical techniques
can be used (i.e. in this paper a nonlinear least-squares method is used). In
addition to the derivation of this technique, the method is validated in two
scenarios and the results are compared to those obtained by the general purpose
optimal control software, GPOPS-II. In general, the proposed technique produces
solutions of $\mathcal{O}(10^{-10})$. Additionally, for the proposed test
cases, it is reported that each individual TFC-based inner-loop iteration
converges within 6 iterations, each iteration exhibiting a computational time
between 72 and 81 milliseconds within the MATLAB legacy implementation.
Consequently, the proposed methodology is potentially suitable for on-board
generation of optimal trajectories in real-time.
