ABSTRACTIn 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.
Numerical robustness of single-layer method with Fourier basis for multiple obstacle acoustic scattering in homogeneous media
