JOURNAL OF SPACECRAFT AND ROCKETS Vol. 51, No. 6, November–December 2014
Statistical Impact Prediction of Decaying Objects A. L. A. B. Ronse∗ and E. Mooij† Delft University of Technology, 2629 HS Delft, The Netherlands
Downloaded by TECHNISCHE UNIVERSITEIT DELFT on December 2, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.A32832
DOI: 10.2514/1.A32832 This paper describes a method for statistically predicting the impact time and location of an orbital object under 200 km altitude. It also investigates the influence of the chief parameter uncertainties on the statistical dispersion. Unlike the main methods used nowadays, which derive drag parameters from historical state data, a shape model is used to acquire aerodynamic coefficients in the various flow regimes. As a test object, the Delta-K rocket stage is chosen, due to the high data availability for validating the tool. Using a Monte Carlo sequence, multiple six-degrees-offreedom simulations are executed, in which the initial state and the density output of the atmosphere model are varied within their uncertainty windows. By kernel density estimation, the resulting data are used to derive a probability density function of the impact time and infer a ground track expressing impact probability. In a comparative study with the Tracking and Impact Predictions published by the United States Space Surveillance Network, the method’s performance is tested. In this comparison, a decrease in impact window size is observed, while maintaining the reliability. Moreover, by a variance-based sensitivity analysis, the uncertainty in the density model is identified as the prime contributor to footprint dispersion and it is shown that knowledge of the rotational state can be critical in decreasing impact windows.
Nomenclature a a B CD CL Cm Cn Cp CS Ct Di D−i DT e g h i Ji;0 Kn L M∞ m RE r Si Sref STi s T Tw
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
m∕s2
acceleration vector, semimajor axis, m ballistic coefficient, m2 ∕kg drag coefficient lift coefficient aerodynamic moment coefficient around y axis aerodynamic moment coefficient around z axis pressure coefficient side force coefficient aerodynamic shear coefficient variance contribution of parameter i variance contribution of all terms excluding i total variance eccentricity gravitational acceleration, m∕s2 altitude, m inclination, rad zonal harmonic gravity coefficient Knudsen number characteristic flowfield dimension, m freestream Mach number mass, kg Earth radius, m radial distance from the Earth’s center, m first-order sensitivity index reference surface, m2 total-order sensitivity index molecular speed ratio orbital period, s wall temperature, K
T∞ t VA x, y, z α α0 β γ θ θω
= = = = = = = = = =
λ μE ρ σ σN σT τ ϕω Ω ω ω0
= = = = = = = = = = =
exospheric temperature, K time, s aerodynamic velocity, m∕s Cartesian coordinates, m angle of attack, rad initial angle of attack, rad angle of sideslip, rad heat capacity ratio local inclination angle (aerodynamics), rad angle between rotation axis’ xy projection and x axis, rad mean molecular free path, m Earth gravitational parameter, m3 ∕s2 atmospheric density, kg∕m3 bank angle, rad normal momentum accommodation coefficient tangential momentum accommodation coefficient true anomaly, rad angle between rotation axis and xy plane, rad right ascension of the ascending node, rad rotational velocity, rad∕s; argument of perigee, rad initial rotational velocity, rad∕s
I.
U
Introduction
NCONTROLLED space debris reentries have occurred on a frequent basis since the first launch vehicles placed objects in Earth’s orbit. In the case of massive objects containing components of high-temperature resistance, the reentry may cause structural, environmental, and safety issues on the Earth’s surface [1]. For taking timely measures in these occasions, reentry and impact predictions are a valuable asset. Other applications of these predictions are found in the avoidance of false-alarm scenarios for missile-warning centers, preparative efforts for the determination of end-of-life deorbit maneuvers, and the postimpact search for debris fragments due to their scientific value or hazardous nature. Various institutes are engaged in the estimation of when and where a decaying object is expected to reenter and impact using (semi-) analytical or numerical propagation and prediction methods, including the U.S. Space Surveillance Network [2] and ESA [3]. In these fields, it is widely known that there are large uncertainties in some of the parameters used in the prediction process, especially related to the decelerations caused by aerodynamic phenomena. For the final reentry phase, advanced tools exist to analyze breakup and survivability characteristics and the resulting fragment dispersion based on object composition and shape [4]. Moreover, statistical
Presented as Paper 2013-4682 at the Atmospheric Flight Mechanics Conference, Boston, MA, 19–22 August 2013; received 15 September 2013; revision received 20 February 2014; accepted for publication 4 March 2014; published online 18 June 2014. Copyright © 2014 by Alexander Ronse and Erwin Mooij. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-6794/14 and $10.00 in correspondence with the CCC. *M.Sc. Space Engineering, Faculty of Aerospace Engineering, Kluyverweg 1;
[email protected]. † Assistant Professor, Astrodynamics and Space Missions, Faculty of Aerospace Engineering, Kluyverweg 1;
[email protected]. Associate Fellow AIAA. 1797
Statistical Impact Prediction of Decaying Objects A.L.A.B. Ronse1 and E. Mooij2 Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 1, 2629 HS Delft, The Netherlands
This paper describes a method for statistically predicting the impact time and location of an orbital object under 200 km altitude. It also investigates the influence of the chief parameter uncertainties on the statistical dispersion. Unlike the main methods used nowadays, which derive drag parameters from historical state data, a shape model is used to acquire aerodynamic coefficients in the various flow regimes. As a test object, the Delta-K rocket stage is chosen, due to the high data availability for validating the tool. Using a Monte-Carlo sequence, multiple 6-degrees-of-freedom simulations are executed, in which the initial state and the density output of the atmosphere model are varied within their uncertainty windows. By kernel density estimation, the resulting data are used to derive a probability density function of the impact time, and infer a ground-track expressing impact probability. In a comparative study with the Tracking and Impact Predictions published by the United States Space Surveillance Network, the method’s performance is tested. In this comparison, a decrease in impact window size is observed, while maintaining the reliability. Moreover, by a variance-based sensitivity analysis, the uncertainty in the density model is identified as the prime contributor to footprint dispersion and it is shown that knowledge of the rotational state can be critical in decreasing impact windows.
1 2
MSc. Space Engineering,
[email protected]. Assistant Professor, section Astrodynamics and Space Missions,
[email protected], Associate Fellow AIAA.
1
I.
Introduction
Uncontrolled space debris re-entries have occurred on a frequent basis since the first launch vehicles placed objects in Earth orbit. In the case of massive objects containing components of high-temperature resistance, the re-entry may cause structural, environmental and safety issues on the Earth’s surface [1]. For taking timely measures in these occasions, re-entry and impact predictions are a valuable asset. Other applications of these predictions are found in the avoidance of false-alarm scenarios for missile-warning centers, preparative efforts for the determination of endof-life de-orbit maneuvers and the post-impact search for debris fragments due to their scientific value or hazardous nature. Various institutes are engaged in the estimation of when and where a decaying object is expected to re-enter and impact using (semi-)analytical or numerical propagation and prediction methods, including the U.S. Space Surveillance Network [2] and the European Space Agency [3]. In these fields, it is widely known that there are large uncertainties in some of the parameters used in the prediction process, especially related to the decelerations caused by aerodynamic phenomena. For the final re-entry phase, advanced tools exist to analyse break-up and survivability characteristics and the resulting fragment dispersion based on object composition and shape [4]. Moreover, statistical methods have shown the effects of re-entry point state errors and variation of break-up altitude [5]. Prior to re-entry, in the orbital decay phase, another class of prediction methods exist, which derive the descent rate and drag properties of objects from multiple previous state observations using a fitting process [6, 7] or average values based on the assumption of a spherical or tumbling cylindrical shape [8, 9]. The shortcomings of these approaches can be found in the fact that they do not consider potential fluctuations in the drag term due to attitude variations and the re-entry windows they provide do not contain any statistical properties. This leads to large uncertainty margins and erroneous re-entry predictions when drag characteristics change due to attitude change. An example is found in the decay of UARS in September 2011, in which a strong evolution of the ballistic parameter was seen during the last 48 hours of its orbital lifetime [10]. In this paper, a new method for decay and impact prediction is presented, which enables statistical footprint estimation for uncontrolled decaying objects during the last days of their descent. It
2
uses a simplified shape description of the considered decaying object for determining its aerodynamic coefficients, allowing 6 degrees of freedom simulations and studying the effect of attitude evolution. By individually taking into account the principal parameter uncertainties and using a Monte Carlo sequence, a statistically varying footprint can be obtained. Moreover, by a variance-based sensitivity analysis, the dispersive effect of the various input parameters are studied. Section II discusses some background information and theory on which the developed method relies. Next, Section III explains how the impact prediction model is constructed and Section IV continues with a short description of the main verification methods and the validation of the tool using true data. This is followed by a discussion of the results in Section V, including a study of the method’s performance compared to existing approaches and the outcome of the sensitivity analysis, as well as some interesting phenomena deduced from the statistical footprints and the sensitivity data. Finally, Section VI summarizes the developed method and its results, and provides conclusions.
II. A.
Theoretical Background
Existing prediction tools and their limitations
The current impact prediction tools generally base their estimates of an object’s drag properties on multiple observations of the translational states. An important source of these data is provided by the Space Surveillance Network (SSN), which publishes the orbital properties of many objects through its Two-Line Element (TLE) catalogue [26]. A recurring element in the existing methods as described in [6–9] is the use of a drag parameter, which is either constant, linearly varying or, in the most elaborate models, dependent on aerodynamic flow properties, described by the Knudsen and Mach number. Drag changes due to attitude variations are not taken into account and assumed to either not occur due to a constant attitude or spherically symmetrical shape, or to ’average out’ due to a sufficiently high rotation rate. Many survivable objects, however, do have varying drag properties as a function of attitude and may change attitude during their decay, due to aerodynamic moments or active attitude corrections. Moreover,
[26] http://www.space-track.org
3
complex rotational dynamics may lead to temporal peaks or dips in the drag properties, having an important effect on the re-entry point. As a clarifying example, in Figure 1 the evolution of the drag coefficient (averaged every 20 minutes) is shown for a simulated rocket stage in the last ∼37 hours of its orbital decay. Here, the frontal surface of the propellant tank is used as reference surface for the aerodynamic coefficients (Sref = 2.27 m2 ). Notice that around 24 hours before re-entry, during ∼40 minutes the drag coefficient, CD , is lower than 5, while between 9 and 7 hours before re-entry, CD approaches 8. These variations may lead to unexpected trajectory evolution and may thus form an important element in short-term re-entry and impact predictions.
8
D
C [−]
6 4 2 0
−35
−30
−25
−20 −15 Time before re−entry [hr]
−10
−5
0
Fig. 1: Evolution of the drag coefficient of a modelled Delta-K rocket body.
Like all applications involving detection, modeling and simulations, impact prediction is a field containing uncertainties. Existing prediction tools take these into account by the introduction of an impact window, a period of time in which the orbital object is expected to decay. The boundaries of this window are generally determined by adding/subtracting a percentage of the remaining lifetime to the estimated impact point, or else by separate simulations, taking into account the cumulative ’worst-case’ effect of offsets in parameters determining the trajectory. For example, for the lower window-boundary (earliest re-entry possibility), the density output of the atmosphere model is adjusted upwards and the ballistic coefficient is chosen to correspond with maximum drag conditions. For the upper boundary, the inverse logic is applied. However, by only looking at the conditions defining the window boundaries, no data can be deduced on the probability distribution within the formed window. Moreover, due to the combined insertion of multiple uncertainties, their individual effects cannot be studied, rendering a sensitivity analysis in terms of contributing 4
parameters impossible. Based on the above reasoning, for enabling a statistical description of impact windows and the investigation of the individual effect of certain parameter uncertainties, the developed impact and prediction method differs from ’conventional’ approaches in the following ways: i) to achieve more details on the effect of attitude uncertainties, aerodynamic properties are deduced from a simplified shape model and are dependent on the flow properties as well as the object’s attitude; ii) for realistic simulations of the drag properties due to attitude evolution, coupling between the translational and rotational state is taken into account. Thus, 6-degrees of freedom (6-DOF) dynamics are simulated; iii) uncertainties in the translational state, rotational state and density output of the atmosphere model are all separately accounted for, thus enabling a sensitivity analysis to determine their individual effect on the resulting footprint dispersion; iv) due to the availability of more computational means than during the development of most of the conventional impact prediction tools, a more rigorous approach can be applied, based on repetitive simulations using varying values for the model uncertainties. This enables (a) the generation of statistical impact windows using a Monte Carlo method and (b) a study of the contributions of parameter uncertainties on footprint dispersion using a variance-based sensitivity analysis.
B.
Object selection
For testing the method and studying true decay cases, it was desirable to model a representative existing object. Having frequently decayed in the past years, the second stage of the Delta 2, the Delta-K rocket body, was chosen. Fragments from this vehicle, including the stainless steel propellant tank, combustion chamber and titanium pressurizer spheres shown in Figure 2, have been occasionally found and studied [11]. Moreover, due to its survivable properties, every decay of the Delta-K is monitored and predicted using a numerical technique by the SSN, for generating socalled Tracking and Impact Predictions (TIPs). This feature allows for a comparison of the results with an independent prediction tool.
5
Fig. 2: Images of recovered Delta II second stage debris. From left to right: propellant tank, thrust chamber and helium pressure tank. Image courtesy: NASA.
C.
Orbital object tracking
An important factor in impact prediction is the availability of measurements of an object’s state at certain epochs prior to re-entry. As decaying objects are often inactive and not capable of measuring/communicating their position and velocity, we have to rely on ground-based measurements for determining these. The SSN maintains an accurate, widely used and well-structured catalog of the present space debris environment. The detection of Earth-orbiting objects is performed on a frequent basis by a widespread network of radar and optical systems, both dedicated and nondedicated, capable of detecting all objects larger than ∼10 cm in Low Earth Orbit (LEO) [27]. This forms an ideal source for (3-DOF) state data of the decaying objects of interest. For further analysis, the SSN measurements are presented in the form of TLE-sets, a widely used format for the description of trajectories of bodies orbiting our planet. For international satellite re-entry prediction campaigns, the Inter-Agency Space Debris Committee (IADC) also utilizes the TLE-format to exchange orbital data of decaying objects [10].
D.
Accelerations
To predict the dynamics of an object, we should know the accelerations it is subjected to. The magnitude of the major accelerations in the 100 to 300 km altitude band are shown in Figure 3,
[27] http://celestrak.com/columns/v04n01/
6
300 GM J2,0
Altitude [km]
250
J3,0 J4,0
200
Drag (high) Drag (low) Moon Sun SRP
150
100 −8 10
−6
−4
10
−2
10 10 Acceleration [m/s2]
0
10
2
10
Fig. 3: Maximum accelerations due to various forces in the 100-300 km altitude band.
computed using theory by Montenbruck and Gill [12]. For the third-body and geopotential perturbations maxima are shown, which in reality only occur at specific conditions and thus limited periods of time. Moreover, contrary to atmospheric drag, most of these effects are periodic. However, taking into account worst-case conditions, it can be concluded that below 200 km altitude, the accelerations on an object are governed by the Earth’s central and oblate gravity terms and atmospheric drag. It can be seen that the latter increases with decreasing altitude and that two curves are depicted, for low and high solar activity. This variation is caused by the changeable properties of the Earth’s atmosphere under these conditions and is an important factor in decay prediction.
1.
Gravitational acceleration
The Cartesian components of gravitational acceleration, including the effect of the J2,0 oblateness term, are given by: µ 1+ r2 µ =− 2 1+ r µ =− 2 1+ r
gx,R = − gy,R gz,R
3 R2 z2 J2,0 3E xR 1 − 5 R2 2 r r 2 2 3 RE zR J2,0 3 yR 1 − 5 2 2 r r 3 R2 z2 J2,0 3E zR 3 − 5 R2 2 r r
(1)
where the subscript R denotes the definition of the vector in the Earth-fixed rotating reference frame and r denotes the radial distance from the Earth center. 7
2.
Aerodynamic acceleration
Aerodynamic forces are the main cause of orbital decay and are thus important to model accurately. The aerodynamic acceleration vector is expressed by:
aA
CD 1 = CS 2 CL
Sref 2 m ρVA
(2)
where the subscript A represents the expression of a vector in the aerodynamic reference frame, ρ is the atmospheric density, m the object’s mass and VA the airspeed. CD , CS and CL are the aerodynamic coefficients in drag, side and lift direction respectively and Sref is the reference area with respect to which the coefficients were determined. Although Equation (2) seems a short and simple relation, the determination of the atmospheric density and aerodynamic coefficients is not so straightforward. Density is retrieved from an atmosphere model, while the coefficients are dependent on flow properties and the object’s rotational dynamics. These phenomena are described in the subsequent sections.
E.
Atmosphere model
Static atmosphere models such as, e.g., the US1976 standard atmosphere are not accurate enough for impact prediction applications, as they do not take into account the changeable atmospheric conditions due to solar and geomagnetic activity fluctuations, nor do they account for temporal day-night or semi-annual variations. More complex semi-empirical models, which are based on satellite decay observations and drag measurements of the past decades, do take these effects into account. One of the most recent models of this type is the NRLMSISE-00 model, released in 2000 by the U.S. Naval Research Laboratory [13]. As it covers the full ground-to-space altitude regime and contains not only a description of the density and temperature, but also data on molecular composition (necessary for the computation of flow characteristics), it is the chosen atmosphere model for this study. Being part of the input parameters of the model, the solar and geomagnetic activity indices are retrieved from space weather data published by Celestrak [14].
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F.
Flow regimes
As previously mentioned in Section II A, the aerodynamic coefficients are dependent on certain flow characteristics. A decaying object crosses different aerodynamic flow regimes during it descent. This process is characterized by a gradual decrease in Knudsen number, Kn, defined as the ratio between the mean molecular free path, λ, and the characteristic flowfield dimension, L [15]. Based on the number density and mass density output of the atmosphere model, the Knudsen number at every position can be computed. When Kn 1, free-molecular flow prevails, and the aerodynamic effects are characterized by individual molecules exchanging momentum with the object’s wall, while intermolecular collisions are very rare. For Kn 1, however, such intermolecular collisions occur much more frequently, and the macroscopic properties of the gas can be considered to vary continuously. The regime in between these flow fields is known as transitional flow, in which the assumptions made for both of the above regimes lose validity. To get an idea of the flow field experienced throughout the last days of the decay trajectory, Figure 4 shows the trajectory resulting from a 3-DOF simulation of an object in a circular orbit at initial altitude of 200 km, with characteristic length, L = 6 m and a (constant) ballistic coefficient of B = 0.0163 m2 /kg. The simulation has been done for moderate solar conditions. The boundaries for the three flow regimes are also shown. Note that, in terms of time, the object resides in the free-molecular flow regime the longest; ca. 6 hours before its re-entry, it enters the transitional flow regime, when its orbital altitude has lowered to about 155 km. The pure continuum regime is only reached after the re-entry, when the object reaches an altitude of ca. 85 km. For the means of decay prediction, the transitional and free-molecular flow regimes are thus the most relevant. However, as in the developed model the aerodynamic coefficients in the transitional regime are determined using a bridging function between the free molecular and continuum solutions, the aerodynamic flow relations in both the free-molecular and (hypersonic) continuum regime are described.
1.
Free-molecular flow
In the free-molecular flow regime, under the assumption of the absence of inter-molecular collisions, the pressure and shear force on an elemental area inclined at angle θ to the flow are computed
9
2
10
Free Molecular regime
0
10
Transitional regime
−2
Kn [−]
10
−4
10
Continuum regime
−6
10
−8
10
0
10
20 Time [hr]
30
40
0
50
100 150 Altitude [km]
200
Fig. 4: Knudsen number as a function of time and altitude.
by: # r 2 2 2 − σN σN Tw √ s sin θ + e−s sin θ 2 T∞ π " # r √ 1 1 T σ w N + 2 (2 − σN ) + s2 sin2 θ + πs sin θ [1 + erf(s sin θ)] s 2 2 T∞
(3)
i σT cos θ h −s2 sin2 θ √ √ e + πs sin θ[1 + erf(s sin θ)] s π
(4)
1 Cp = 2 s
Ct =
"
where q∞ represents the dynamic pressure, s the molecular speed ratio,
Tw T∞
the ratio between wall
and free-stream temperature and σN and σT are the normal and tangential momentum accommodation coefficients. It has been empirically shown that, for general engineering surfaces as used on satellites and rocket bodies, the diffuse reflection model best describes the molecule-surface interaction and σN ≈ σT ≈ 1. Finally, the molecular speed ratio, s, is related to the Mach number according to: r s = M∞
2.
γ 2
(5)
Continuum flow
An object re-entering from a decaying orbit and surviving re-entry initially travels at hypersonic speed, subsequently decelerating to supersonic and in some cases subsonic velocities before impact. 10
However, for the means of re-entry/impact predictions at the timescale we are assessing (initial conditions up to 200 km altitude), only the hypersonic regime plays a notable roll in the process of statistical impact prediction. In this regime, a widely used set of methods for the estimation of the pressure distribution around an object are the local-inclination methods. These methods assume that the flow characteristics along a certain object are dependent only on the inclination each surface element has with respect to the incoming flow. Although providing merely a simplified model of the flow, they result in a fast analysis with reasonable accuracy [16]. The thin shock-layer close to the surface, a characteristic for hypersonic flows, enables the approximation of the flow properties by Newtonian theory, in which the incoming airflow is diverted in a tangential direction when reaching the surface, thereby losing all of its normal momentum. In the case of blunt bodies, to which most parts of survivable debris such as rocket bodies and satellites can be counted, the pressure coefficient of a surface can best be described by a Mach-dependent version, known as modified Newtonian flow [17]: Cp = Cpmax sin2 θ
(6)
where Cpmax is the maximum value of the pressure coefficient at a stagnation point behind a normal shock wave:
Cpmax
2 = 2 γM∞
"
2 (γ + 1)2 M∞ 2 4γM∞ − 2(γ − 1)
γ/(γ−1)
# 2 1 − γ + 2γM∞ −1 γ+1
(7)
Note that Equation (6) is only applicable for positively inclined surfaces. It is known that for high Mach numbers, the contribution of leeward surface elements to the total pressure coefficient becomes negligible [16]. Therefore, the Cp of all leeward elements is assumed 0. Note that when using Equation 6 for the determination of aerodynamic coefficients, only inviscid force contributions are considered. Due to their limited effect on the pressure distribution [16], the effect of viscous phenomena are thus not taken into account, greatly simplifying the analysis.
11
3.
Transition flow
For the transitional flow regime, a bridging function is used between the coefficients found by applying above theory. The following empirically determined function is used [18]: F (Kn) = sin2
3 + log10 Kn π 8
(8)
This function can be used to determine the force and moment coefficients when 10−3 < Kn < 10, according to: CX − CXcont = F (Kn) C¯X = CXFM − CXcont
(9)
where the subscript FM denotes the respective free-molecular coefficient and cont represents the continuum variant.
G.
Statistical methods
For the generation of statistical footprints and the sensitivity analysis, the developed method uses some statistical concepts which are explained in this section.
1.
Monte Carlo method
To account for the uncertainties in some of the chief model and state parameters, a Monte Carlo method is used. This approach relies on repetitive simulations with randomly assigned values of the uncertain parameters within their statistical windows. Simply said, many (e.g., 1000) simulations are done, each having an independent and randomly generated set of values for the unknown or uncertain parameters. For every simulation, a different impact epoch and location results and the full set of output data can be used to infer statistical properties related to the impact prediction.
2.
Extended Fourier Amplitude Sensitivity Test
One of this study’s goals is to determine the main contributing parameters to the prediction uncertainties. This is done by a sensitivity analysis, relating the dispersion in the model’s output to different sources of uncertainty in its inputs. Using the results from this analysis, it is possible to identify the model inputs causing the most significant output uncertainties, and on which focus
12
should thus be put when striving for better impact prediction. To realize this, a variance-based method is used, the Extended Fourier Amplitude Sensitivity Test (EFAST). This method enables the deduction of both first- and total-order indices from a limited set of simulations [19]. Unlike the (pseudo-)random sample generation as used in the Monte Carlo method, EFAST’s sample generation is based on the deliberate variation of parameters across their uncertainty interval at pre-defined frequencies. By performing a Fourier analysis of the model outputs, the measured dispersion can be coupled to variational frequencies of the input parameters, thus deriving the effect of each parameter on the output variance. The first-order sensitivity index Si of a certain parameter is then determined by:
Si =
Di DT
(10)
where Di expresses the variance caused by the uncertainty of parameter i without any interactions with other parameters and DT represents the total variance of the footprint. As noted above, by clever selection of the variational frequencies of the parameters, EFAST also allows for the total sensitivity index, STi , to be computed. This parameter expresses the full contribution of parameter i, including its interactions with other parameters. It is determined by: STi = 1 −
D(−i) DT
(11)
where D(−i) represents the variance caused by all complementary parameters and their interactions.
III.
Impact Prediction Model
This section describes how the theory described in the previous sections is used to construct a tool enabling statistical impact prediction and a parametric study of impact dispersion. First, a general outline of the method is given in Section III A. Next, the model’s major components are described in more detail, starting with the aerodynamic coefficient generation in Section III B. The stochastic sample generation, necessary to account for model and state uncertainties, is described in Section III C. The actual 6-DOF state propagation until impact is described in Section III D. Finally, in Section III E, the logic is explained for using the simulator output to construct statistically varying impact footprints. 13
A.
Model outline
The impact prediction method can be divided into five main components, as shown in Figure 5: Aerodynamic coefficient generation This part of the simulation process uses a simplified shape model to acquire aerodynamic coefficients of the object for specified attitude and Mach windows in the free-molecular and hypersonic flow regimes. It is written in C++ and outputs an ASCII data-file for each flow regime. More details are given in Section III B. Sample generation In this component, the uncertainties in the translational and rotational state, as well as those related to the density output of the atmosphere are incorporated by the generation of samples according to a certain statistical distribution. The sample generation is performed in MATLAB and explained in Section III C Data storage The input samples as well as impact predictions resulting from the simulation, are stored in an SQLite database [28]. This storage method was chosen for its convenient way of arranging the large amount of data, and its easy and fast interface with both the MATLAB and C++ environment, used for the sample generation, state propagation and data analysis components. State propagation This component consists of the numerical integration of an orbital object’s 6-DOF state based on the accelerations and moments it experiences during its decay. This routine is written in C++, due to its high performance for problems of high computational load and the availability of a wide scope of used routines in the C++ based TU Delft Astrodynamics Toolbox (Tudat) [20]. More details on the propagation are provided in Section III D. Statistical analysis The output data are analyzed in MATLAB. This software was chosen for its extensive statistical toolbox and visualization options. A description of the applied methods for the construction of statistical footprints is given in Section III E.
[28] http://www.sqlite.org/about.html
14
Aerodynamic coefficient generators C++
EFAST Monte-Carlo sample sample generator generator MATLAB MATLAB
Aerodynamic Coefficients
Aerodynamic coefficient database ASCII
Samples Sample generation
Statistical analysis
Database SQLite Data storage
Aerodynamic coefficient generation
MATLAB Data analysis
Decay and re-entry simulator C++ Impact prediction
State propagation
Fig. 5: General flow diagram of the software.
B.
Aerodynamic coefficient generation
For computing the translational and rotational accelerations of a tumbling object due to aerodynamic forces and moments, it is necessary to determine its aerodynamic coefficients at different attitudes, flow regimes and Mach numbers. For these means, the object shape is first approximated by a collection of simple shapes and its surface is panelized by a quadrilateral meshing sequence. In case of the modeled Delta-K rocket body, the result is shown in Figure 6. For every panel, the centroid, area and outward normal direction are determined. These parameters are subsequently used in the determination of aerodynamic coefficients. For the computation of the free-molecular coefficients, Equations (3) and (4) can be applied to the separate elements of the panelized surface to determine the shear and pressure distribution along the object’s surface at a certain incident flow direction. Thereby the panel centroid, area and surface normals, deduced from the meshed geometry, are used. The shear direction is determined using the object’s velocity vector and surface normal. As an example, the resulting pressure distributions at two angles of attack are shown in Figure 7 for a free-molecular flow at M = 20. Note that an algorithm is used to determine which panel segments of the object are shadowed by others. These
15
regions can be clearly distinguished behind the rocket body’s mini-skirt and the inside of the nozzle. As incoming molecules do not hit shadowed panels, the pressure coefficient of these is assumed zero. Using similar logic, Equations (6) and (7) can be used to compute the pressure distribution along the whole object’s positively inclined surface in the hypersonic regime. As explained above, the panelized object model is used to generate aerodynamic coefficients at various attitudes and Mach numbers in both the free-moleculear and hypersonic regime. These coefficients are stored in two separate databases which can be used to determine the coefficients in all three flow regimes. The coefficient generation process is based on three independent variables: angle of attack, α, angle of sideslip, β and Mach number, M . As the attitude of an arbitrarily tumbling object is unbounded, α and β are varied across their full range: −π
