Homotopy method for optimization of variable-specific-impulse low-thrust trajectories

“…Taking the high-thrust problem as the initial problem and performing the homotopy on the thrust magnitude to reduce the sensitivity of long flight duration and many revolutions is another common application of homotopy methods [20]. Properly applying these homotopy methods can solve most of the trajectory optimization problems in the two-body problem [21]- [23]. Furthermore, Caillau completed many researches about the trajectory optimization in the CRTBP, including the minimum-time problem and the minimum-fuel problem [24]- [26].…”

Section: Introductionmentioning

confidence: 99%

Practical Homotopy Methods for Finding the Best Minimum-Fuel Transfer in the Circular Restricted Three-Body Problem

Pan

2020

IEEE Access

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Finding a solution to a low-thrust minimum-fuel transfer trajectory optimization problem in the circular restricted three-body problem (CRTBP) is already known to be extremely difficult due to the strong nonlinearity and the discontinuous bang-bang control structure. What is less commonly known is that such a problem can have many local solutions that satisfy all the first-order necessary conditions of optimality. In this paper, the existence of multiple solutions in the CRTBP is fully demonstrated, and a practical technique that involves multiple homotopy procedures is proposed, which can robustly search as many as desired local solutions to determine the best solution with the optimal performance index. Due to the common failure of the general homotopy method in tracking a homotopy path, the parameter bounding fixed-point (PBFB) homotopy is coupled to construct the double-homotopy, which can overcome the difficulties by tracking the discontinuous homotopy path. Furthermore, the PBFB homotopy is capable to achieve multiple solutions that lie on separate homotopy path branches. The minimum-time problem is first solved by the doublehomotopy on the thrust magnitude to infer a reasonable transfer time. Then the auxiliary problem is solved by the double-homotopy and the PBFP homotopy, and multiple local solutions with smooth control profiles are obtained. By tracking homotopy paths starting from these solution points, multiple solutions to the minimum-fuel problem are obtained, and the best solution is eventually identified. Numerical demonstration of the minimum-fuel transfers from a GEO to an Earth-Moon L 1 halo orbit is presented to substantiate the effectiveness of the technique. INDEX TERMS Circular restricted three-body problem, Earth-Moon transfers, homotopy method, trajectory optimization.

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“…The indirect method has been demonstrated to be a promising method to realise ascent trajectory generation of space shuttles, multi-burn launch vehicles (10) and hypersonic air-breathing vehicles (11) . By improving the convergence property through combining the analytical vacuum initial guess technology with the homotopy technology, the solving rate can be improved (9,12) . In order to achieve a highprecision solution of TPBVPs, a large number of discrete intervals are needed in the finite difference method, which will lead to low solving rate and poor reliability.…”

Section: Introductionmentioning

confidence: 99%

Rapid multi-layer method on solving optimal endo-atmospheric trajectory of launch vehicles

Wei

Han

et al. 2019

Aeronaut. j.

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In order to increase the speed, precision and robustness against the engine failure in solving optimal endo-atmospheric ascent trajectory of a launch vehicle, a rapid multi-layer solving method with improved numerical algorithms was proposed. The proposed method is capable of decomposing a large number of intervals into multiple layers with advantageous convergence property. Firstly, the problem of solving optimal endo-atmospheric ascent trajectory, which was subjected to path constraints and terminal constraints, was transformed into a Hamilton Two Point Boundary Value Problem (TPBVP). Then, through the finite difference method and numerical solving algorithm, the Hamilton TPBVP was iteratively solved with fewer initial discrete intervals. The initial values of higher-layer iterations were obtained by interpolating convergent solutions at sparse nodes into the doubly discrete nodes of high layers. The process was repeatedly performed until the solving precision met the requirements. To decrease the calculation load in solving TPBVPs, two improved solving algorithms without and with fewer Jacobian calculations were studied, respectively the Derivative-free Spectral Algorithm for Nonlinear Equations(DF-SANE) combined with the improved derivative-free nonmonotone line search strategy, and the Modified Newton method with a relaxation factor in combination with the Inverse Broyden Quasi-Newton method, denoted as ‘MN-IBQ’. Simulation verifications showed that the multi-layer method had significantly higher solving speed than the single-layer method. For the improved numerical algorithms, the DF-SANE was trapped in the local convergence problem. While using the proposed MN-IBQ can further increase the solving rate. Typical engine failure simulations showed that the multi-layer method with the MN-IBQ algorithm had not only significantly higher solving speed but also stronger robustness, where the traditional single-layer method could not adapt. In addition, the thrust loss tolerance limits for the multi-layer solving method were given for different engine failure times. The results show promising potential of the proposed approach in trajectory online generation and closed-loop guidance of launch vehicles at the endo-atmospheric ascent stage.

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Homotopy method for optimization of variable-specific-impulse low-thrust trajectories

Cited by 33 publications

References 22 publications

Design and optimization of low thrust transfer trajectory for engineering constraints

Design and optimization of low thrust transfer trajectory for engineering constraints

Practical Homotopy Methods for Finding the Best Minimum-Fuel Transfer in the Circular Restricted Three-Body Problem

Rapid multi-layer method on solving optimal endo-atmospheric trajectory of launch vehicles

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