# Initial Guesses for Finite Maneuver Optimization

The solution of finite-maneuver optimization problems using the direct transcription method requires that an initial guess or approximation for the optimization decision variables be supplied to the transcription-optimization system. Astrogator uses Legendre and Radau Pseudospectral global collocation methods (LPM and RPM), a global collocation method, for transcription, and the Sparse Nonlinear Optimizer (SNOPT) package for solving the nonlinear programming problem (NLP) that results from this transcription. Following LPM and RPM, the decision variables are the orbit states and the control functions (thrust attitude by spherical angles or unit vectors) at a set of nodes, and this set may also include the maneuver start/end time(s). Accordingly, an initial guess must be provided to the transcription solver for state-control functions sampled at these nodes. If the initial/final time of the maneuver is free, a guess for this epoch must also be provided. In Astrogator, a user can either import a pre-generated initial guess in the form of an Initial Guess File, or use an existing finite maneuver, as a source for the initial guess. Accordingly, Astrogator offers the following two methods of "seeding" an optimal finite maneuver:

• an Initial Guess File (*.nod), and,
• a finite maneuver

After an initial guess has been provided to Astrogator at a set of generic points following either of the above mechanisms, the "raw" guess data is internally resampled/interpolated at the quadrature nodes (e.g. the Legendre-Gauss-Lobatto nodes for LPM and Legendre-Gauss-Radau nodes for RPM) before numerical optimization is initiated.

The quality of an initial guess is critical in determining the convergence performance of a gradient-based numerical optimization library such as SNOPT. A "good" initial guess is one which is "close" to a locally optimum solution of the problem under consideration, and/or is near-feasible. See references [1, 2, 3] for details.

[1] Bruce Conway, ed., Spacecraft Trajectory Optimization. Cambridge University Press, Cambridge, England, U.K., 2010., Chap. 1

[2] John T. Betts., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming. Society for Industrial & Applied Mathematics; 2nd edition., 2009., Chaps.1–3.

[3] Jorge Nocedal and Stephen Wright., Numerical Optimization. Springer 2nd edition., 2006.