Analytical Spacecraft Trajectory Optimization

Abstract

This monograph examines the problem of trajectory optimization for spacecraft operating within a Newtonian field. Background information of the problem formulation is provided, including the overall investigation objective. A survey of previous works is provided with regards to optimization by means of indirect and direct methods. A formulation of spacecraft equations of motion is provided following definitions of applicable coordinate systems. Specific methods of optimization are conferred in their numerical form, with most attention given to shooting methods for the reason that it was the dominant method used to obtain research results. Direct optimization through collocation is addressed in terms of Runge-Kutta and trapezoidal methods. The document further addresses the conditions of optimality in numerical form, discussing formulation of a performance index for optimality and then classifying applicable conditions of optimality into either first-order or higher-order. Trajectories constructed are either coplanar, in polar coordinates, or non-coplanar, in spatial (spherical) coordinates. Planar maneuvers are designed by first applying optimality conditions to properly formulate equations of motion and costate equations. The two-point boundary value problem resulting from this methodology is solved for the specific cases of constant thrust, switching thrust (also known as bang-off-bang), and variable specific impulse using numerical methods. Problem types in which singular arcs may occur are addressed using Intermediate Thrust arc segments in the form of Lawden Spirals, for which explicit analytical solutions are derived. Intermediate Thrust trajectories are performed using one, two, or three intermediate thrust arcs, contributing various levels of initial or final orbit definition between elliptical Keplerian orbits. In addition to juxtaposition of performance throughout the various trajectory designs, existence of viable solutions in the case of three Intermediate Thrust arc segments is derived in terms of compulsory terminal orbit conditions. Spatial maneuvers are considered for continuous thrust and Intermediate Thrust trajectories between non-coplanar elliptical orbits following a similar optimality conditions application as that of planar maneuvers. Equations of spacecraft motion are defined and costate equations derived in addition to first integrals of the system and invariant relations. Mass-flow rate of a free time horizon Intermediate Thrust arc is obtained through invariant relations, expressed as a function of current state and Lagrange multipliers. Fuel efficiency of trajectories for non-coplanar maneuvers are compared for the constant thrust and Intermediate Thrust cases, as well as to a case of direct optimization methods. Additionally, an amelioration to Lawden Spirals is offered through formulation of explicit state expressions for Intermediate Thrust arcs of non-coplanar operations.