The numerical solution of Optimal Control Problems has received much attention over the past few decades. In particular, direct transcription methods have been studied because of their convergence properties for even relatively poor guesses at the solution. This thesis explores two of these techniques—Legendre-Gauss-Lobatto (LGL) Pseudospectral (PS) Collocation and Multiple Shooting (MS), and draws distinct comparisons between the two, allowing the reader to decide which would be better applied to a particular application. Specifically, comparisons will be made on accuracy, computation time, adjoint estimation, and storage requirements. It will be shown that the most distinct advantage for LGL PS collocation and MS methods will lie in computation time and storage requirements, respectively, using a nonlinear interior-point method described in this thesis.