Analytical solutions to three-dimensional hypersonic gliding trajectory over rotating Earth

In this paper, high-precision analytical solutions are developed for 3-dimensional (3-D) hypersonic gliding trajectory over the rotating and spherical Earth using the regular perturbation method. First, an Auxiliary Geocentric Rotating (AGR) frame fixed on the rotating Earth is created based on a generalized equator, which is defined as the great circle in the plane containing the initial position and velocity vectors of the hypersonic vehicle, and then a new entry dynamics model is established based on the generalized longitude and latitude. As the rotational angular velocity of Earth is no longer just along the z-axis of the AGR frame, the new model becomes much more complex than the conventional one. However, the benefit of this model is that the AGR frame caters to the trajectory characteristic that the downrange is much greater than the crossrange. Thus the model is conducive to being linearized. By simplifying the model skillfully, a reduced-order system is obtained with energy as independent variable and capable of predicting the downrange, crossrange and heading angle of 3-D trajectory. However, in this system, the crossrange and heading angle are still coupled seriously. To cope with this, the system is decomposed into a Taylor series by the regular perturbation method and approximated by truncating the series after the first two terms. The remaining two terms are linear time-varying (LTV) sub-systems and can be solved by the spectral-decomposition-based method. In the derivation process, we find and prove that some complex formulae related to Earth's rotation keep almost constant since the glider flies near the generalized equator. The simulation results demonstrate that the new analytical solutions have high accuracy and require a small amount of calculation.

Automated summary

This paper presents high-precision analytical solutions for hypersonic gliding trajectories over the rotating and spherical Earth using the regular perturbation method. The new model established based on the generalized equator and latitude captures the trajectory characteristic well. By skillfully simplifying the model, a reduced-order system is obtained capable of predicting the 3-D trajectory, although the coupling between crossrange and heading angle remains a challenge to be overcome.