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 ($A_D$) and the ratio of the horizontal component of the lift to the drag ($L_{Dz}$), which are important for governing the 3D trajectory, emerge as key parameters of the dynamical equations. Then, $A_D$ and $L_{Dz}$ 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.

为了在未来设计基于阻力的预测校正器进入制导，开发了表示为有限项切比雪夫级数的新分析公式，用于快速预测三维 (3D) 高超声速滑翔轨迹 (HGT)，方法是提出用于求解高度非线性入口动力学模型的类牛顿方法。具体来说，首先，通过适当简化原始动力学模型来开发新的降阶动力学模型，使得阻力加速度（$A_{丁}$) 以及升力的水平分量与阻力的比值 (${\mathrm{大号}}_{丁z}$)，这对于控制 3D 轨迹很重要，成为动力学方程的关键参数。然后，$A_{丁}$和${\mathrm{大号}}_{丁z}$被计划为速度的多项式。然而，由于其高度非线性，简化模型仍然不易求解。为了解决这个困难，提出了一种类似牛顿的方法，将简化模型转换为线性微分递归方程（LDRE）。通过反复评估 LDRE，可以生成 3D HGT 的一系列近似值，其极限是该简化模型指定的轨迹。事实上，由于类牛顿法收敛速度快，LDREs 只需求解两次即可达到高精度。通过使用切比雪夫级数来逼近 LDREs 中出现的一些复杂项，LDREs 变得解析可解。结果，获得了 3D HGT 的近似解析解。仿真结果表明，解析解需要在笔记本电脑上花费不到 1 毫秒的时间，并且误差小于 4%。计算效率是机载预测校正器制导的基础。