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This paper provides an effective approach for the prediction and estimation of space debris due to a vehicle breakup during uncontrolled reentry. For an advanced analysis of the time evolution of space debris dispersion, new efficient computational approaches are proposed. A time evolution of the dispersion of space pieces from a breakup event to the ground impact time is represented in terms of covariance ellipsoids, and in this paper, two covariance propagation methods are introduced. First, a derivative-free statistical linear regression method using the unscented transformation is utilized for performing a covariance propagation. Second, a novel Gaussian moment-matching method is proposed to compute the estimation of the covariance of a debris dispersion by using a Gauss-Hermite cubature-based numerical integration approach. Compared to a linearized covariance propagation method such as the Lyapunov covariance equation, the newly proposed Gauss-Hermite cubature-based covariance computation approach could provide high flexibilities in terms of effectively representing an initial debris dispersion and also precisely computing the time evolution of the covariance matrices by utilizing a larger set of sigma points representing debris components. In addition, we also carry out a parametric study in order to analyze the effects on the accuracy of the covariance propagation due to modeling uncertainties. The effectiveness of the newly proposed statistical linear regression method and the Gauss-Hermite computational approach is demonstrated by carrying out various simulations.

In the past 50 years, over 16,000 metric tons of man-made space objects and unexpected asteroids have entered the Earth’s atmosphere. Although most of them burn while entering the atmosphere, some of the debris have survived to impact the Earth’s surface and do expose a risk to people and property [

Methodology for designing a reentry trajectory of an uncontrolled space object.

In order to compensate for the drawbacks of the Lyapunov-based estimation of the debris dispersion, alternative solutions could be employed by increasing the order of the Taylor-series expansion of the nonlinear system or by using an advanced numerical technique [

The main contributions of this paper are twofold. First, a precise and effective Gaussian cubature transformation-based [

The remainder of this paper is organized as follows: Section

Let

ENU (east-north-up) coordinate system for reentry equations of motion.

The governing equation of motion for an uncontrolled space object entering into the atmosphere perturbed by the atmospheric drag uncertainty but ignoring the lifting force can be described by the following equations with position

Now, after defining an augmented state vector

It is assumed that the initial nominal state of a reentering space vehicle right before the breakup is given by propagating an orbital decay reentry trajectory until 78 km in ECEF (Earth-centered-Earth-fixed). After the coordinate transformation from the ECEF to ENU frame, the initial position of a breakup event is given by geometric altitude

Then, the nonlinear translational equations of motion in (

In this section, two new approaches for the estimation of debris dispersion due to a space vehicle breakup are proposed. The estimation of debris dispersion is represented in terms of a covariance propagation which describes probabilistic ellipsoids from a nominal reentry trajectory. First, a statistical linear regression using the unscented transformation is introduced to compute the time evolution of the covariance information. Second, a Gaussian moment-matching method which calculates the first and second moments of the probabilistic distribution of the debris dispersion by using a Gauss-Hermite cubature numerical approximation is proposed. Figure

(a) Lyapunov equation-based covariance propagation. (b) Statistical linear regression-based covariance propagation method—unscented transformation approach.

In general, statistical linear error propagation is more accurate than the error propagation by first-order Taylor-series approximation. For a statistically linearized error propagation, consider a nonlinear mapping with a set of weighted sigma points

Then, the first- and second-order statistics of the transformed points, mean, and covariance, are computed from the sigma point

Before applying a statistical linear regression method, it is necessary to transform the continuous nonlinear differential equation in (

As for the prediction step for the next time

Even though the statistical linear regression (SLR) using the unscented transformation approach could represent the propagation of the debris dispersion due to a breakup event, the SLR-based approach estimates the debris dispersion with the second-moment covariance using a finite small number of sigma points. The statistical regression approach is not a truly global approximation and it does not work well with nearly singular covariances, that is, nearly deterministic systems with small covariances. For compensating the drawbacks, it is necessary to consider a more realistic approach which can precisely capture the initial debris dispersion and also take into account the neglected nonlinearities from the statistical linear regression. In this section, the Gaussian cubature transformation approach [

In Gauss-Hermite cubature integration rule, the lattice points in dimension ^{n}

Consider the transformation of the state

The debris dispersion can be modeled with the first

Illustration of a fifth-order Gauss-Hermite cubature-based approximation to a nonlinear transformation. The covariance of the true distribution is presented by the dashed line and the solid line is the approximation [

Original

Transformed

The integral in (

The unit sigma points

Also, the multidimensional unit sigma points can be given as Cartesian product of the one-dimensional unit sigma points

It is noted that the number of sigma points required for

Based on the previous Gauss-Hermite cubature integration approach, an additive form of the multidimensional Gauss-Hermite cubature integral technique is used to represent the covariance prediction of a debris dispersion. It is assumed that the posterior density function

As for the prediction step for the next time

The multidimensional Gaussian-Hermite cubature-based covariance prediction approach which represents the debris dispersion and the distribution in time can be interpreted as a special form of a Monte-Carlo integration approach.

In this section, first, various parametric studies are carried out to analyze the effects of uncertainties in the atmospheric density and drag coefficient along with wind effects. Then, the performance of the proposed estimation techniques are investigated by comparing two covariance propagation methods; the unscented transformation approach and the Gauss-Hermite cubature integration method.

Atmospheric drag most strongly influences the motion of reentry objects near the Earth. When it comes to determining accurately atmospheric drag, the values of atmospheric density and drag coefficient are the main factors to the trajectory in the sense that the information on the specific shape and quantity of objects are generally unknown. In light of this reason, the different values of the drag coefficient are selected to see the how the drag coefficient affects the reentry point and the probabilistic ellipsoid of debris dispersion caused by the breakup event in this study. In this study, four different atmospheric density models including CIRA-72, Exponential model, U.S. standard model, and NRLMSISE-00 are used to analyze the effects of the atmospheric density model onto the reentry trajectory. Figure

Density profiles for density models.

Wind profiles for different altitudes.

In order to analyze the effects of the density model and the drag coefficient, two simulation examples are investigated. The first example considers the lifetime prediction of uncontrolled space objects with different atmospheric density models and drag coefficients. The satellite under consideration has the following orbit parameters:

Decay of altitude with different atmospheric models.

Table

Terminal data of lifetime prediction with different density models.

Density model | Reentry latitude | Reentry longitude | Reentry time | STK LTP tool |
---|---|---|---|---|

CIRA-72 | −38.936° | −345.39° | Aug. 26 (01:15) | Aug. 25 (18:21) |

U.S. standard | −27.306° | −224.34° | Aug. 28 (15:44) | Aug. 27 (09:02) |

NRLMSISE-00 | −8.302° | −319.91° | Aug. 20 (05:31) | Aug. 19 (03:22) |

Decay of altitude with different drag coefficients.

On the other hand, a second example is illustrated to show the effects of different density models and drag coefficient onto the debris nominal trajectory as well as the time evolution of the debris dispersion from 78 km at the breakup event to impact ground instance. For the simulation study, the number of 200 samples is drawn to model the initial debris breakup dispersion with the assumption of a Gaussian distribution. It is assumed that the initial breakup state is given by

The values of each latitude and longitude are taken with

Figures

Time evolution of probability ellipsoid with different density models.

Time evolution of probability ellipsoid with different drag coefficients.

Impact footprint (density models).

Impact footprint (drag coefficients).

Footprint statistics for various atmospheric density models.

Density model | Impact latitude | Impact longitude | Impact area |
---|---|---|---|

CIRA-72 | −3.007° | −240.2° | 1522 km^{2} |

U.S. standard | −27.306° | −224.34° | 619.2 km^{2} |

NRLMSISE-00 | −8.302° | −319.91° | 913.8 km^{2} |

Footprint statistics for various drag coefficients.

Drag coefficient | Impact latitude | Impact longitude | Impact area |
---|---|---|---|

CD = 1.5 | 12.23° | −137.3° | 1093 km^{2} |

CD = 2.0 | −7.452° | −347.3° | 916.5 km^{2} |

CD = 2.5 | −11.96° | −118.3° | 547.3 km^{2} |

In this section, a detailed analysis of the estimation of debris dispersion due to the breakup event during the reentry is made by using one of the proposed covariance propagation methods, the unscented transformation technique. Note that the positional variation at the breakup instant is small enough to be neglected but the variation of the debris velocity becomes relatively big. Therefore, it is assumed that an initial breakup dispersion could be generated by adding an incremental speed _{0} is remodeled with a Gaussian distribution before covariance propagation for the time evolution of the debris dispersion in (

Note that the initial breakup dispersion generated with an incremental speed

Initial estimate of debris dispersion.

Since the probabilistic distribution of the debris dispersion is Gaussian, the center of the ellipsoid becomes the breakup point and the dispersion is represented by an ellipsoid. Uncertainty due to unknown and neglected acceleration terms is characterized by using the process noise covariance, that is,

It is assumed that the breakup altitude is 78 km and the initial breakup position is located at

Breakup point and impact footprint.

After the generation of the initial sigma points, the covariance is propagated by using (

Figure

Dispersion probability ellipsoids for a sequence of time and impact footprint on the ground.

This phenomenon is verified in Figure

Magnitude of positional covariances in each direction (

For a better analysis of the first few minutes, a detailed view is illustrated in Figure

Initial dispersion of ellipsoids after the breakup event within 100 seconds.

The simulation is made with different ballistic coefficients using the empirical density and wind models. Figure

Plot of the empirical ballistic coefficient as a function of flight time [

Simulated eastern and northern wind profiles for different altitudes.

Effects of the wind to the time evolution of the decay of the debris altitude.

In this section, the performance of the proposed covariance methods for estimating the debris dispersion due to the reentry breakup is investigated in terms of the time evolution of the positional probabilistic ellipsoid. The covariance propagation methods include the unscented transformation-based statistical linear regression and the Gauss-Hermite cubature-based numerical integration methods described in Section

For the simulation test, it is also assumed that the breakup altitude is 78 km and the initial breakup position is located at

Figure

Performance comparison between unscented transformation and Gauss-Hermite cubature methods in nominal trajectory prediction and ground impact footprint.

Figures

Time evolution of debris dispersion ellipsoids and impact footprint on the ground using the statistical linear regression with the unscented transformation method.

Time evolution of debris dispersion ellipsoids and impact footprint on the ground using the Gauss-Hermite cubature integration method.

Comparison of the magnitude of positional covariances in each direction (

In this paper, for advanced analysis of the time evolution of space debris dispersion due to the breakup during reentry, two effective computational approaches were proposed to estimate the statistical distribution of the debris dispersion. The estimated debris dispersion is represented by using the prediction of positional probability ellipsoids for the visualization of the results. First, the time evolution of the covariance propagation was computed by using a statistical linear regression using the unscented transformation. Second, a novel covariance estimation technique was proposed by utilizing the Gaussian moment-matching method with a Gauss-Hermite cubature-based numerical integration approach. Compared to other covariance propagation methods, the newly proposed Gauss-Hermite cubature-based covariance computation could not only represent more exact initial dispersion, but also precisely calculate the time evolution of the debris dispersion with a large set of sigma points representing debris components. The Gauss-Hermite cubature approach does not require any linearization of the nonlinear translation equations of motion of a reentering debris, which leads to precisely taking into account nonlinearities exposed on the motion of a reentering debris. Furthermore, we also carried out a parametric study in order to analyze the effects on the accuracy of the covariance propagation due to modeling uncertainties. For the detailed parametric analysis, four types of density models and drag coefficients were used in the computation of the lifetime prediction and the covariance computation. In the simulation studies, it was shown that the newly proposed statistical linear regression method and the Gauss-Hermite computational approach are in close agreement to each other, but the Gauss-Hermite covariance propagation method provides a more precise estimate of the debris dispersion of the breakup fragments.

The authors declare that they have no conflicts of interest.

This study was supported by the Korea Astronomy and Space Science Institute (KASI) through the research project “Study on Precise Trajectory and Trajectory Uncertainty Prediction of Space Objects for Ground Impact Risk Assessment” (Project no. 2017-1-854-01) supervised by the Ministry of Science and ICT.

The video materials show the simulation result of the time evolution of the multiple breakup dispersion during the reentry phase using unscented transform-based covariance propagation. This simulation helps to intuitively understand one example of reentry debris dispersion with the 3-dimensional ellipsoid.