Neighboring optimal guidance using higher-order dynamics approximation with application to orbital injection problem

Neighboring optimal guidance, a method to obtain a suboptimal guidance law by approximately solving the first-order necessary conditions based on a nominal trajectory, is widely used in the aerospace field due to its high computational efficiency and low resource usage. For more advanced scenarios, the existing methods still have a problem that the guidance accuracy and optimality will seriously degrade when the actual state largely deviates from the nominal trajectory. This is mainly caused by the approximate description of the first-order conditions in terms of total flight time and nonlinear constraints. To address this problem, a higher-order neighboring optimal guidance method is proposed. First, a novel total flight time updating strategy, together with a normalized time scale, is presented that transforms the optimal problem with free total flight time into a more tractable optimal problem with fixed total flight time. Then, using the vector partial derivative method, a higher-order approximation is adopted, instead of the first-order approximation, to accurately describe the nonlinear dynamical and terminal constraints, thus obtaining a polynomially constrained quadratic optimal problem. Finally, to numerically solve the polynomially constrained quadratic optimal problem, a Newton-type iterative algorithm based on the orthogonal decomposition is designed. Through the iterative solution within each guidance period, the corrections to control quantities and total flight time are generated. The proposed method is applied to a launch vehicle orbital injection problem, and simulation results show that it achieves high accuracy of orbital injection and optimality of performance index.