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