design space pruning heuristics and global optimization method for ...

more robust convergence. The solution is generally less sensitive to the initial guesses
and those initial guesses are more physically intuitive, which make direct methods
preferable for implementing within an automated global optimization scheme.
Differential dynamic programming (DDP) also parameterizes the control
variables, providing a large convergence domain and decreasing the sensitivity to poor
initial guesses. As compared to direct methods, DDP is less sensitive to the high
dimensionality of the low-thrust trajectory optimization problem as it transforms the large
problem into a succession of low dimensional sub-problems. Quadratic programming is
then used on each resulting quadratic sub-problem to solve for controls that improve the
trajectory locally. The states and objective function are then calculated forward in time
using the updated controls, and the process is repeated until the problem has converged.
One disadvantage of DDP is that it is most effective for smooth unconstrained problems.
Low-thrust problems, however, tend to include numerous constraints and can be highly
non-smooth. In recent work, Lantoine and Russell have modified the traditional DDP
algorithm to create a hybrid differential dynamic programming algorithm that addresses
some of the weaknesses of DDP. The hybrid approach uses first- and second-order state
transition matrices to calculate the partial derivates required for optimization, and
combines DDP with NLP techniques to increase its robustness and efficiency. 222324
Finally, there also exist hybrid methods which numerically integrate the Euler-
Lagrange equations and control the spacecraft based on the primer vector.14 ,25
As in the
direct method, hybrid methods solve a nonlinear programming problem, but with the
Lagrange multipliers making up part of the parameter vector while maximizing or
minimizing some cost function. Hybrid methods search numerically for the set of
parameters that extremize the cost function, while explicitly satisfying kinematic
boundary constraints. According the work by Gao and Kleuver, the advantages of hybrid
trajectory optimization methods include a significant reduction in the design space and
7