Indirect Optimization of Spiral Trajectories

Indirect optimization is used to compute minimum propellant spiral escapes and captures. A two-step estimation process generates accurate estimates of the Lagrange multipliers. The first step, an adjoint control transformation, converts the thrust unit direction vector to the actual multipliers that control the trajectory. Next, curve fits are matched with the values of the initial multipliers from spirals found with the adjoint control transformation and used to extrapolate the multipliers for longer spirals. Spherical rather than Cartesian coordinates are used because the spherical multipliers evolve in a well-behaved fashion, allowing the accurate extrapolation of the multipliers. Long-duration Earth escapes are presented for spirals as long as 150 days. Solutions are also presented with control limits on the thrust or specific impulse. Additionally, a transformation is developed that converts the spherical multipliers to the corresponding Cartesian multipliers and vice versa. Another transformation is developed to convert the optimal multipliers for two-dimensional, planar equatorial spirals into the optimal multipliers for planar-inclined orbits in either Cartesian or spherical coordinates in which the initial orbit may have arbitrary values for inclination, right ascension of the ascending node, and argument of periapsis.