Chebyshev-series solutions for nonlinear systems with hypersonic gliding trajectory example

Abstract

In order to design a drag-based predictor-corrector entry guidance in the future, new analytical formulae, expressed as finite-term Chebyshev series, are developed for fast predicting a three-dimensional (3D) Hypersonic Gliding Trajectory (HGT) by proposing a Newton-like method to solve a highly nonlinear entry dynamics model. Specifically, first, a new reduced-order dynamics model is developed by properly simplifying the original dynamics model such that the drag acceleration (AD) and the ratio of the horizontal component of the lift to the drag (LDz), which are important for governing the 3D trajectory, emerge as key parameters of the dynamical equations. Then, AD and LDz are planned as polynomials of speed. However, the simplified model is still not easy to solve due to its high nonlinearity. To deal with the difficulty, a Newton-like method is proposed to transform the simplified model into Linear Differential Recurrence Equations (LDREs). By evaluating the LDREs repeatedly, a sequence of approximations of the 3D HGT can be generated, the limit of which is the trajectory specified by that simplified model. In fact, due to the fast convergence speed of the Newton-like method, the LDREs need be solved only twice to achieve high accuracy. By using Chebyshev series to approximate some complex terms appearing in the LDREs, the LDREs become analytically solvable. As a result, the approximate analytical solutions to the 3D HGT are obtained. Simulation results show that the analytical solutions need to consume less than 1 ms of time on a laptop computer and have errors of less than 4 percentage. The computational efficiency is fundamental for onboard predictor-corrector guidance.