Research

The research aims to improve uncertainty propagation of space debris for conjunction analysis and re-entry predictions. One part is the estimation and modeling of uncertainties in the initial state and environment models. Another concerns the actual propagation of the uncertainties and the subsequent analysis of the models and parameters.

### Publications

**Geul, J**., Mooij, E. & Noomen, R (2018). Analysis of Uncertainties and Modeling in Short-Term Re-Entry Predictions.

Journal of Guidance, Control, and Dynamics, Volume 41, Number 6, pp. 1276–1289, doi:10.2514/1.G003258

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Satellite re-entry predictions are used to determine the time and location of impacts of decaying objects. These predictions are complicated by uncertainties in the initial state and environment models, and the complex evolution of the attitude. Typically, the aerodynamic and error propagation are done in a simplistic fashion. Full six-degrees-of-freedom modeling and attitude control is proposed for studying the historic re-entry case of the Gravity field and steady-state Ocean Circulation Explorer satellite. Improved error modeling and estimation of the initial state and atmospheric density are introduced for both global positioning system and two-line elements states. A sensitivity analysis is performed to identify the driving parameters for several models and epochs. The predictions are compared against tracking and impact predictions, and predictions by the European Space Agency Space Debris Office. The performed predictions are consistently closer to the true decay epoch for several starting epochs, while providing narrower windows than other predictions with higher confidence.

- Hoogendoorn, R., Mooij, E. &
**Geul, J.** (2018). Uncertainty propagation for statistical impact prediction of space debris.

Advances in Space Research, Volume 61, Issue 1, pp. 167–181, doi:10.1016/j.asr.2017.10.009

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Predictions of the impact time and location of space debris in a decaying trajectory are highly influenced by uncertainties. The traditional Monte Carlo (MC) method can be used to perform accurate statistical impact predictions, but requires a large computational effort. A method is investigated that directly propagates a Probability Density Function (PDF) in time, which has the potential to obtain more accurate results with less computational effort. The decaying trajectory of Delta-K rocket stages was used to test the methods using a six degrees-of-freedom state model. The PDF of the state of the body was propagated in time to obtain impact-time distributions. This Direct PDF Propagation (DPP) method results in a multi-dimensional scattered dataset of the PDF of the state, which is highly challenging to process. No accurate results could be obtained, because of the structure of the DPP data and the high dimensionality. Therefore, the DPP method is less suitable for practical uncontrolled entry problems and the traditional MC method remains superior. Additionally, the MC method was used with two improved uncertainty models to obtain impact-time distributions, which were validated using observations of true impacts. For one of the two uncertainty models, statistically more valid impact-time distributions were obtained than in previous research.

Keywords: uncertainty propagation; statistical impact prediction; space debris; Liouville equation.

**Geul, J**., Mooij, E. & Noomen, R (2017). TLE uncertainty estimation using robust weighted differencing.

Advances in Space Research, Volume 59, Issue 10, pp. 2522–2535, doi:10.1016/j.asr.2017.02.038

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Accurate knowledge of satellite orbit errors is essential for many types of analyses. Unfortunately, for two-line elements (TLEs) this is not available. This paper presents a weighted differencing method using robust least-squares regression for estimating many important error characteristics. The method is applied to both classic and enhanced TLEs, compared to previous implementations, and validated using Global Positioning System (GPS) solutions for the GOCE satellite in Low-Earth Orbit (LEO), prior to its re-entry. The method is found to be more accurate than previous TLE differencing efforts in estimating initial uncertainty, as well as error growth. The method also proves more reliable and requires no data filtering (such as outlier removal). Sensitivity analysis shows a strong relationship between argument of latitude and covariance (standard deviations and correlations), which the method is able to approximate. Overall, the method proves accurate, computationally fast, and robust, and is applicable to any object in the satellite catalogue (SATCAT).

Keywords: space debris, error analysis, SGP4 accuracy, space situational awareness (SSA), conjunction analysis, pairwise differencing.

**Geul, J**., Mooij, E. & Noomen, R (2017). GOCE Statistical Re-entry Predictions.

7th European Conference on Space Debris, Volume 7, Issue 1.

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Satellite re-entry predictions are used to determine the time and location of impacts of decaying objects. Large uncertainties result from unknowns in the initial state and environment models. The complex evolution of the attitude further complicates these predictions, especially for slender bodies. Typically, predictions only propagate a point-mass and assume a static error window. Full six degrees-of-freedom (6DOF) statistical re-entry predictions of ESA’s Gravity field and steady-state Ocean Circulation Explorer (GOCE) are proposed. Improved error models for the initial state and atmospheric density are introduced. The uncertainty parameters are estimated using Global Positioning System orbit solutions. The predictions are compared against Tracking and Impact Predictions (TIPs) and predictions by the ESA/ESOC Space Debris Office. The 6DOF predictions are consistently closer to the true decay epoch for several starting epochs, while providing narrower windows than TIPs.

Keywords: space debris; re-entry predictions; error propagation; rigid-body dynamics.

**Geul, J**., Mooij, E. & Noomen, R (2017). Modelling and Assessment of the Current and Future Space Surveillance Network.

7th European Conference on Space Debris, Volume 7, Issue 1.

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Two-Line Elements (TLEs) and Tracking and Impact Predictions (TIPs) resulting from the Space Surveillance Network (SSN) are important for many Space Situational Awareness (SSA) activities. The network consists of many different radar and optical stations, contributing in either a dedicated, collateral or contributing fashion. For the majority of objects in low-Earth orbit (LEO) phased-array radar (PAR) stations are essential for maintaining the satellite catalog (SATCAT). A survey of the current state of the SSN and a methodology for simulating the network is presented. The current state is compared to a future scenario with the new Space Fence System (SFS) included. The SSN is simulated as a collection of current and hypothetical dedicated and collateral PARs. The location and coverage of each sensor is investigated and modelled. Observations are generated for a test rocket body in LEO. TLEs are estimated from the simulated measurements using the Simplified General Perturbations (SGP4) model. The sensitivity of the fit residuals and propagated accuracy with respect to eccentricity, inclination, and number of observations and sensors is analysed. The orbit-determination solution is found to be most sensitive to the eccentricity, and number of observations and sensors involved. The new SFS improves the solution, especially for lower inclinations.

Keywords: space debris; space surveillance; phased-array radar; satellite catalog (SATCAT); orbit determination; two-line Elements (TLEs).

**Geul, J**., Mooij, E. & Noomen, R (2016). Regularised Methods for High-Efficiency Propagation.

Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Volume 156, August 9-13, 2015, Vail, CO, USA, Paper AAS 15-697, pp. 4105-4124.

Published 2016, ISSN: 0065-3438, ISBN: 978-0-87703-629-6.

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Although regularised propagation methods have a good performance (accuracy versus evaluations), they suffer from a number of practical difficulties, such as propagation to a fixed time, making them ill-suited for practical applications. Several methods that address these limitations are proposed, thoroughly discussed, and analysed on diverse test cases. Dromo outperforms the conventional propagation methods significantly. It is shown that regularised methods, through some adaptations, can be successfully applied to different orbit problems. The proposed method is recommended especially for computationally demanding problems.

Keywords: regularization, time transformation, numerical integration, multi-step methods, Hermite interpolation, root finding, space debris.

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