A Fast Second-Order Algorithm for Preliminary Design of Low-Thrust Trajectories
- Paper number
IAC-08.C1.2.5
- Author
Mr. Gregory Lantoine, Georgia Institute of Technology, United States
- Coauthor
Dr. Ryan Russell, Georgia Institue of Technology, United States
- Year
2008
- Abstract
Low-thrust propulsion systems are becoming increasingly considered for future space missions because propellant is used more efficiently. However the optimization of the resulting trajectories is numerically challenging since the thrust is operated for significant periods of the mission time, which leads to large set of independent variables to optimize.
To overcome this issue and reduce dramatically the number of variables, we use a well-proven technique by modeling continuous thrusting as a series of impulses. The trajectory is subdivided into segments with an impulsive ΔV at the beginning of each segment. To get a good approximation of low-thrust propulsion systems, the magnitude of each impulse is constrained by the maximum amount of ΔV that could be accumulated over the duration of the segment by the corresponding low-thrust engine. To reduce computational time, the propagation between impulses is done by using a Lambert two-body solver.
However, the resulting constrained, nonlinear optimization problem is still challenging to solve. To further enhance parameter reduction, instead of solving the problem by a classical NLP solver, we use an alternative approach based on a second-order Differential Dynamic Programming (DDP) technique - an extension of Bellman Dynamic Programming theory. Being second-order, this algorithm exhibits quadratic convergence if sufficiently close to the optimal trajectory. Other advantages are that the solution has by nature a feedback component and the curse of dimensionality is kept under control by concentrating calculations in the neighborhood of a nominal trajectory. Each iteration can be divided in two parts: the backward and forward sweeps. In the forward sweep, the states and objective value of the system are calculated. In the backward sweep, an update to the current thrust control policy is determined through the computations of the sensitivities of the objective function with respect to small variations of the control sequence. These expressions are based on the two-body analytical first-order and second-order state transition matrices along all of the segments. This leads to a tremendous reduction in computational time compared to classical optimization methods that integrate a set of equations to get those derivatives.
Several different types of trajectories are optimized using our approach to demonstrate its feasibility and competitiveness. The results are compared to those from BNDSCO, a trajectory optimization program using an indirect method. Our method provides similar results while being fast, easy of use and showing small convergence sensitivity. It is therefore highly valued for preliminary design where large trade spaces have to be assessed rapidly.
- Abstract document
- Manuscript document
IAC-08.C1.2.5.pdf (🔒 authorized access only).
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