This paper demonstrates the use of polynomial chaos expansions for the nonlinear, non-Gaussian propagation of orbit state uncertainty. Using linear expansions in tensor products of univariate orthogonal polynomial bases, polynomial chaos expansions approximate the stochastic solution of the ordinary differential equation describing the propagated orbit, and include information on covariance, higher moments, and the spatial density of possible solutions. Results presented in this paper use non-intrusive, i.e., sampling-based, methods in combination with either least-squares regression or pseudospectral collocation to estimate the polynomial chaos expansion coefficients at any future point in time. Such methods allow for the usage of existing orbit propagators. Samples based on sun-synchronous and Molniya orbit scenarios are propagated for up to ten days using two-body and higher-fidelity force models. Tests demonstrate that the presented methods require the propagation of orders of magnitude fewer samples ...

Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos

Stochastic optimal control for aircraft conflict resolution under wind uncertainty

Polynomial chaos expansion and Gaussian mixture models are combined in a hybrid fashion to propagate state uncertainty for spacecraft with initial Gaussian errors. Polynomial chaos expansion models uncertainty by performing an expansion using orthogonal polynomials. The accuracy of polynomial chaos expansion for a given problem can be improved by increasing the order of the orthogonal polynomial expansion. The number of terms in the orthogonal polynomial expansion increases factorially with dimensionality of the problem, thereby reducing the effectiveness of the polynomial chaos expansion approach for problems of moderately high dimensionality. This paper shows a combination of Gaussian mixture model and polynomial chaos expansion, Gaussian mixture model–polynomial chaos expansion as an alternative form of the multi-element polynomial chaos expansion. Gaussian mixture model–polynomial chaos expansion reduces the overall order required to reach a desired accuracy. The initial distribution is converted to a...

Spacecraft Uncertainty Propagation Using Gaussian Mixture Models and Polynomial Chaos Expansions

Uncertainty in aircraft trajectory planning and prediction generates major challenges for the future air traffic management system. Therefore, understanding and managing uncertainty will be necessa...

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Robust Aircraft Trajectory Planning Under Wind Uncertainty Using Optimal Control

Robust trajectory optimization using polynomial chaos and convex optimization

The development of algorithms for aircraft robust dynamic optimization considering uncertainties (for example, trajectory optimization) is relatively limited compared to aircraft robust static optimization (for example, configuration shape optimization). In this paper, an approach for dynamic optimization considering uncertainties is developed and applied to robust aircraft trajectory optimization. In the present approach, the nonintrusive polynomial chaos expansion scheme is employed to convert a robust trajectory optimization problem with stochastic ordinary differential equations into an equivalent deterministic trajectory optimization problem with deterministic ordinary differential equations. Two computational strategies for trajectory optimization considering uncertainties are compared. The performance of the developed method is studied by considering a classical deterministic trajectory optimization problem of supersonic aircraft short-time climb with uncertainties in the aerodynamic data. Detailed...

Chen Gao

Papers

Aircraft Robust Trajectory Optimization Using Nonintrusive Polynomial Chaos