## Abstract

The paper is presenting a newly developed modular toolbox named Strathclyde Mechanical and Aerospace Research Toolbox for Uncertainty Quantification (SMART-UQ) that implements a collection of intrusive and non intrusive techniques for polynomial approximation and propagation of uncertainties. Non intrusive methods build the polynomial approximation of the uncertain states through sampling of the uncertain parameters space and interpolation. Intrusive methods redefine operators in the states model and perform the states evaluation according to the newly defined operators.

The main advantage of non intrusive methods is their range of applicability since the model is treated as a black box hence no regularity is required. On the other hand, they suffer from the curse of dimensionality when the number of required sample points increases. Intrusive techniques are able to overcome this limitation since they have lower computational cost than their corresponding non intrusive counterpart. Nevertheless, intrusive methods are harder to implement and cannot treat the model as a black box. Moreover intrusive methods are able to propagate nonlinear regions of uncertainties while non intrusive methods rely on hypercubes sampling.

The most widely known intrusive method for uncertainty propagation in orbital dynamics is Taylor Differential Algebra. The same idea has been generalized to Tchebycheff and Newton polynomial basis because of their fast uniform convergence with relaxed continuity and smoothness requirements. However the SMART-UQ toolbox has been designed in a flexible way to allow further extension of the intrusive and non-intrusive methods to other basis.

The Generalized Intrusive Polynomial Expansion (GIPE) approach, implemented in the toolbox and presented here in the paper, expands the uncertain quantities in a polynomial series in the chosen basis and propagates them through the dynamics using a multivariate polynomial algebra. Hence the operations that usually are performed in the space of real numbers are now performed in the algebra of polynomials therefore a polynomial representation of the uncertain states is available at each integration step. To improve the computational complexity of the method, arithmetic operations are performed in the monomial basis. Therefore a transformation between the chosen basis and the monomial basis is performed after the expansion of the elementary functions.

Non intrusive methods have been implemented for a set of sampling techniques (Halton, Sobol, Latin Hypercube) for interpolation in the complete polynomial basis as well as on sparse grid for a reduced set of basis.

In the paper the different intrusive and non intrusive techniques integrated in SMART-UQ will be presented together with the architectural design of the toolbox. Test cases on propagation of uncertainties in space dynamics with the corresponding intrusive and non intrusive approaches will be discussed in terms of computational cost and accuracy.

The main advantage of non intrusive methods is their range of applicability since the model is treated as a black box hence no regularity is required. On the other hand, they suffer from the curse of dimensionality when the number of required sample points increases. Intrusive techniques are able to overcome this limitation since they have lower computational cost than their corresponding non intrusive counterpart. Nevertheless, intrusive methods are harder to implement and cannot treat the model as a black box. Moreover intrusive methods are able to propagate nonlinear regions of uncertainties while non intrusive methods rely on hypercubes sampling.

The most widely known intrusive method for uncertainty propagation in orbital dynamics is Taylor Differential Algebra. The same idea has been generalized to Tchebycheff and Newton polynomial basis because of their fast uniform convergence with relaxed continuity and smoothness requirements. However the SMART-UQ toolbox has been designed in a flexible way to allow further extension of the intrusive and non-intrusive methods to other basis.

The Generalized Intrusive Polynomial Expansion (GIPE) approach, implemented in the toolbox and presented here in the paper, expands the uncertain quantities in a polynomial series in the chosen basis and propagates them through the dynamics using a multivariate polynomial algebra. Hence the operations that usually are performed in the space of real numbers are now performed in the algebra of polynomials therefore a polynomial representation of the uncertain states is available at each integration step. To improve the computational complexity of the method, arithmetic operations are performed in the monomial basis. Therefore a transformation between the chosen basis and the monomial basis is performed after the expansion of the elementary functions.

Non intrusive methods have been implemented for a set of sampling techniques (Halton, Sobol, Latin Hypercube) for interpolation in the complete polynomial basis as well as on sparse grid for a reduced set of basis.

In the paper the different intrusive and non intrusive techniques integrated in SMART-UQ will be presented together with the architectural design of the toolbox. Test cases on propagation of uncertainties in space dynamics with the corresponding intrusive and non intrusive approaches will be discussed in terms of computational cost and accuracy.

Original language | English |
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Number of pages | 9 |

Publication status | Published - 16 Mar 2016 |

Event | 6th International Conference on Astrodynamics Tools and Techniques - Darmstadt, Germany Duration: 14 Mar 2016 → 17 Mar 2016 |

### Conference

Conference | 6th International Conference on Astrodynamics Tools and Techniques |
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Country | Germany |

City | Darmstadt |

Period | 14/03/16 → 17/03/16 |

## Keywords

- uncertainty quantification
- uncertainty propagation
- polynomial algebra
- polynomial interpolation