Hybrid Gaussian Mixture Splitting Techniques for Uncertainty Propagation in Nonlinear Dynamics

T HE uncertainty propagation problem has been studied in many applications in astrodynamics, such as orbit determination [1], conjunction assessment [2], planetary reentry [3], and relative motion [4]. An accurate and efficient propagation of the uncertainty of the mentioned applications is crucial for the reliability of simulation results in associated missions. Analytical or semi-analytical nonlinear methods have been developed for uncertainty propagation. A review of uncertainty propagation in orbital mechanics is given by Luo and Yang [5]. The Monte Carlo (MC) technique is the baseline method for nonlinear uncertainty propagation. It provides true uncertainty propagation results but is computationally intensive. To reduce the computational load of MC, many nonlinear methods have been presented, such as unscented transformation (UT) [6], state transition tensor (STT) [7], differential algebra (DA) [8], Gaussian mixture model (GMM) [9], and the hybrid methods of GMM-UT [10] and GMM-STT [4]. UT and STT are used to nonlinearly propagate the first two statistical moments (i.e., the mean and the covariance matrix), and high-order statistics [11,12]. To propagate higher-order statistical moments and the probability density function (PDF), the GMM method and all its related hybrid methods (i.e., GMM-UT, GMM-STT) can be used. For high-dimensional and nonlinear problems, the introduction of domain splitting [13] can improve the accuracy of the uncertainty propagation techniques by subdividing the entire initial uncertainty domain into smaller subdomains. For example, Wittig et al. [13] introduce the automatic domain splitting (ADS) technique into the DA algorithm to accurately propagate large uncertainty sets in highly nonlinear orbit dynamics.