Propagation and Reconstruction of Reentry Uncertainties Using Continuity Equation and Simplicial Interpolation

Abstract

This Paper proposes a continuum-based approach for the propagation of uncertainties in the initial conditions and parameters for the analysis and prediction of spacecraft reentries. Using the continuity equation together with the reentry dynamics, the joint probability distribution of the uncertainties is propagated in time for specific sampled points. At each time instant, the joint probability distribution function is then reconstructed from the scattered data using a gradient-enhanced linear interpolation based on a simplicial representation of the state space. Uncertainties in the initial conditions at reentry and in the ballistic coefficient for three representative test cases are considered: a three-state and a six-state steep Earth reentry and a six-state unguided lifting entry at Mars. The Paper shows the comparison of the proposed method with Monte Carlo-based techniques in terms of quality of the obtained marginal distributions and runtime as a function of the number of samples used.

Comparison of continuity equation and Gaussian mixture model for long-term density propagation using semi-analytical methods

“…given the actual expressions of v α i , the result of the density n(α 1 , •••, α m , t) can be obtained by numerical integration together with the state variables α i . Apart from the numerical way to solve the density evolution equation (Trisolini and Colombo 2021), analytical solutions can be found for specific problems (McInnes 2000;Colombo and McInnes 2011b;Letizia et al 2015).…”

“…The performance of DEE is mainly constrained by the performance of the linear interpolation and the number of initial samples. The problem of obtaining better linearly interpolated results for the case of highly deformed and elongated phase space distribution (Trisolini and Colombo 2021) will be given into a deeper insight in the future work. For GMM-UT, the splitting of the initial Gaussian distribution is performed at the initial time along the solar angle direction.…”

“…The accuracy of DEE with linear interpolation mainly depends on the dynamical nonlinearity and the performance of linear interpolation. To improve the performance of linear interpolation when the phase space is highly deformed and elongated, an improved linear interpolation method considering alpha shape (Trisolini and Colombo 2021) can be studied in the future work. The accuracy of GMM-UT mainly depends on the dynamical nonlinearity, the splitting number of sub-Gaussians (i.e., theoretically, as the number of splitting increases, a GMM can approach the true non-Gaussian density distribution), the splitting direction (i.e., whether the splitting is done along the most nonlinear direction of the dynamical system), whether the splitting is done within various directions at the initial time , whether the splitting is done during the propagation by detecting the nonlinearity of the state uncertainty (DeMars et al 2013;Romano 2020), and the performance of UT for nonlinear propagation of the first two statistical moments of each sub-Gaussian.…”

“…The uncertainty propagation problem has been analysed in many research directions in astrodynamics, such as orbit determination (Lopez-Jimenez et al 2020;Jia and Xin 2020), relative motion (Yang et al 2018), planetary re-entry (Halder and Bhattacharya 2011;Jiang and Li 2019;Trisolini and Colombo 2021). A review of uncertainty propagation methods in orbit mechanics is given in Luo and Yang (2017).…”

A multi-segment alpha shape-based continuum method for long-term density propagation with bifurcation

“…In [10], the authors classify the methods into two categories: parametric (where one evolves the statistical moments) and nonparametric (where one approximates or directly propagates the evolution equation of the Probability Density Function (PDF)). Representative nonparametric methods are Gaussian Mixture Model (GMM)-based hybrid methods, such as Gaussian Mixture Model-Unscented Transformation (GMM-UT) [6], Density Evolution Equation (DEE, or continuity equation) [18,22,24,25]. The former approximates the non-Gaussian PDF at any time using the weighted sum of the first two statistical moments of the Gaussians.…”