NONLINEAR MAPPING OF UNCERTAINTIES: A DIFFERENTIAL ALGEBRAIC APPROACH

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1 NONLINEAR MAPPING OF UNCERTAINTIES: A DIFFERENTIAL ALGEBRAIC APPROACH R. Armellin 1, P. DiLizia 1, F. Bernelli-Zazzera 1, and M. Berz 2 1 Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La masa 34, 2156 Milano, Italy 2 Department o Physics and Astronomy, Michigan State University, East Lansing, Michigan, 48824, USA ABSTRACT A method or the nonlinear propagation o uncertainties in celestial mechanics based on dierential algebra is presented. The arbitrary order Taylor expansion o the low o ordinary dierential equations with respect to the initial condition delivered by dierential algebra is exploited to implement an accurate and computationally eicient Monte Carlo algorithm, in which thousands o pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter o asteroid Apophis with our planet in 229. To this aim, we irst compute the high order Taylor expansion o Apophis close encounter distance rom the Earth by means o map inversion and composition; then we run the proposed Monte Carlo algorithm to perorm the statistical analysis. Key words: Uncertainties Propagation; Monte Carlo Simulation; Apophis Close Encounter; Dierential Algebra. The propagation o uncertainties in orbital mechanics is usually addressed by linear propagation models [1, 7, 17] or nonlinear Monte Carlo simulations [14]. The main advantage o the linear methods is the simpliication o the problem, but their accuracy drops o or highly nonlinear systems and/or long time propagations. On the other hand, Monte Carlo simulations provide true trajectory statistics, but are computationally intensive. The tools currently used or the robust detection and prediction o planetary encounters and potential impacts o Near Earth Objects (NEO) are based on these techinques [8, 9, 15], and thus suer the same limitations. The eect o the coordinate system on the propagated statistics is analyzed by Junkins et al. [18, 19] and used to develop an alternative approach to orbit uncertainty propagation. However, this method is based on a linear assumption and thus cannot map nonlinearities. An alternate way to analyze trajectory statistics by incorporating higher-order Taylor series terms that describe localized nonlinear motion is proposed by Park and Scheeres [2]. Their approach is based on proving the integral invariance o the probability density unction via solutions o the Fokker Planck equations or diusionless systems, and by combining this result with the nonlinear state propagation to derive an analytic representation o the nonlinear uncertainty propagation. As a result, the method enables the nonlinear mapping o Gaussian statistics, bypassing the drawbacks o Monte Carlo simulations. However, it is limited to systems derived rom a single potential. 1. INTRODUCTION Dierential algebraic (DA) techniques are proposed as a valuable tool to develop an alternative approach to tackle the previous tasks. Dierential algebra supplies the tools to compute the derivatives o unctions within a computer environment [4, 5, 6]. More speciically, by substituting the classical implementation o real algebra with the implementation o a new algebra o Taylor polynomials, any unction o n variables is expanded into its Taylor polynomial up to an arbitrary order k. This has an important consequence when the numerical integra-

2 tion o an ordinary dierential equation (ODE) is perormed by means o an arbitrary integration scheme. Any explicit integration scheme is based on algebraic operations, involving the evaluation o the ODE right hand side at several integration points. Thereore, starting rom the DA representation o the initial condition and carrying out all the evaluations in the DA ramework, the low o an ODE is obtained at each step as its Taylor expansion in the initial condition [1]. The availability o such high order expansions is exploited when problems with uncertain initial conditions have to be analyzed. As the accuracy o the Taylor expansion can be kept arbitrarily high by adjusting the expansion order, the approach o classical Monte Carlo simulations can be enhanced by replacing thousands o integrations with evaluations o the Taylor expansion o the low. As a result, the computational time reduces considerably without any signiicant loss in accuracy. The algorithm is applied to the prediction o Apophis planetary encounter and potential impact, taking into account its measurement uncertainties. The availability o high order maps in space and time, and intrinsic tools or their inversion, are exploited to reduce the computation o the close encounter distance (CED) rom the Earth o all the asteroids belonging to the initial uncertainty cloud (commonly reerred to as virtual asteroids [16]) to the simple evaluation o polynomials. Similar techniques exploiting high order Taylor expansions o the low o ODE and their inverses obtained with DA techniques have already been eiciently utilized in beam physics. Two noticeable applications are the reconstruction o trajectories in particle spectrographs together with the reconstructive correction o residual aberrations [2], and the end-to-end simulations o ragment separators [11]. This paper presents an application to celestial mechanics. 2. DA INTEGRATION OF APOPHIS DY- NAMICS 2.1. Dynamical Models The study o the motion o a NEO in the Solar System with an accuracy suicient to predict collisions requires the inclusion o various relativistic corrections to the well-known Newtonian orces based on the Kepler s orce law. Speciically, the ull equation o motion in the Solar System including the relevant relativistic eects are considered. It is assumed that the object we are integrating is aected by the gravitational attraction o k bodies, but has no gravitational eect on them; i.e., we are adopting the restricted (n + 1)-body problem approximation. The positions, velocities, and accelerations o the k bodies are considered as given values, computed by cubic spline interpolations o data retrieved rom HORIZONS Web-Interace ( These interpolations are necessary as in the DA ramework the use o external code is not permitted. The cubic splines are built so as to keep the maximum error with respect to HORIZONS ephemerides o the order o 1 9 AU and 1 1 AU/day or bodies position and velocity, respectively (see [3] or details). In our integrations k includes the Sun, planets, the Moon, Ceres, Pallas, and Vesta. For planets with moons, with the exception o the Earth, the center o mass o the system is considered. The dynamical model is written in the J2. Ecliptic reerence rame and is commonly reerred to as Standard Dynamical Model [12]. To improve the integration accuracy the dynamics are scaled by Earth semimajor axis and Sun gravitational parameter (i.e., a E = 1 and µ S = Gm s = 1). We must mention that, to obtain a ull understanding o the dynamics o a body in the Solar System, other eects should be taken into account, such as: the orces due to other natural satellites and asteroids, the J 2 (and higher order harmonics o the potential) eect o the Earth and other bodies, Yarkosvsky and solar radiation pressure eects [12]. When the asteroid approaches the Earth, a dierent set o ODE are integrated to avoid cancellation errors associated to repetitive subtraction o Apophis and Earth s position vectors occurring across the lyby pericenter. In this case the equations o motion are written in the J2. Earth- Centered Inertial reerence rame. The same gravitational bodies o the heliocentric phase are considered, whereas relativistic corrections are neglected as their eect during a ast close encounter is negligible. In this phase the dynamics are scaled by the radius o the Earth and by the Earth gravitational parameter (i.e., R E = 1 and µ E = Gm E = 1).

3 2.2. Flow Expansion The high order expansion o the low o ODE can be straightorwardly obtained by evaluating any explicit numerical integration scheme within the DA ramework. The results presented here are obtained by applying a DA-based 8-th order Runge Kutta Fehlberg (RKF78) scheme with absolute and relative tolerance o The integration window is June 18, 29 to April 16, 229, being April 13, 229, the day in which the close approach occurs. The nominal initial state and the associated σ o Apophis, expressed in equinoctial variables p = (a, P 1, P 2, Q 1, Q 2, l), are taken rom the Near Earth Object Dynamic Site (newton.dm.unipi.it/neodys) and summarized in Table 1. Apophis initial condition is initialized y [AU] Earth Apophis x [AU] Figure 1. Apophis heliocentric phase trajectory. 2 x 1-3 Apophis Moon Table 1. Apophis equinoctial variable at 3456 MJD2 (June 18, 29) and associated σ values. Units: a [AU] and l [deg]. y [AU] 1 Nom Value a P P Q Q l as DA variables [p ] = p + 3σδp, where 3σ is used as a scaling actor. These variables are converted into cartesian coordinates using the relations given in Battin [1], evaluated in the DA ramework and then numerically propagated. Note that the solution o the Kepler equation, required or the computation o the eccentric longitude, is carried out by applying the DA-algorithm introduced in Di Lizia et al. [1]. The nominal heliocentric trajectories o Apophis and the Earth are shown in Fig. 1 by the solid and dotted lines, respectively. Figure 2 shows a zoom o Apophis close approach with the Earth in the geocentric reerence rame. It is worth mentioning that the maximum norm o the dierence between the computed trajectory and Apophis HORIZONS ephemerides σ x [AU] x 1-3 Figure 2. Apophis close encounter trajectory. is less than AU. The mismatch is due to all dierent initial conditions, dynamical and ephemeris model, and integration scheme. An analysis on the accuracy o the low expansion is mandatory beore introducing the DAbased Monte Carlo algorithm. Figures 3 and 4 show the maximum position and velocity error o the Taylor representation o the low at the corners o the initial set, with respect to the pointwise integration o the same points. Initial widths o 3, 6, and 9 σ and expansion orders rom 1 to 5 are considered. The expansion error decreases when higher expansion orders are selected and when smaller uncertain sets are considered. The errors tend to decrease exponentially with the expansion order, until reaching a lower limit o approximately [AU] on position and [AU/day] on velocity. It is worth notic-

4 Expansion error on position [AU] σ 6 σ 9 σ Expansion error on velocity [AU/day] σ 6 σ 9 σ Expansion order Figure 3. Accuracy o the Taylor expansion o the low corresponding to dierent expansion orders and initial uncertainty sets: position error Expansion order Figure 4. Accuracy o the Taylor expansion o the low corresponding to dierent expansion orders uncertainty sets: velocity error. ing that a ith order expansion guarantees a gain o approximately three order o magnitude in the low representation with respect to linear methods. This gain can be crucial when impact probability and/or resonant returns are studied. The igure clearly shows that Taylor polynomial accuracy is a unction o both the expansion order and domain width. The drawback or obtaining the Taylor expansion o the low with respect to the initial condition is the computational time to perorm a single integration, as shown in Fig. 5. In this igure the ratio between the computational time o a k-th order DA integration and a single pointwise integration is illustrated, underlining that a 5-th order integration is approximately eight times slower. On the other hand, the availability o the low expansion enables the development o a computationally eicient Monte Carlo method, as described in the next section. 3. DA-BASED MONTE CARLO Within the dynamical models adopted and the chosen integration scheme, the asteroid reerence solution has a close encounter distance rom the Earth center o mass o km at epoch MJD2. In order to evaluate the possibility o an Earth s impact it is necessary to accurately propagate the statistics o the asteroid. The accurate computation o statistics in nonlinear dynamical systems oten relies on Monte Carlo simulations. The algorithmic low o a Monte Carlo simulation is: Ratio o computational time Expansion order Figure 5. DA integration computational time compared to single pointwise integration. 1. Generate random samples based on the statistical distribution o the uncertainty to be propagated. 2. Run a pointwise integration o each sample in the ully nonlinear dynamics. 3. Perorm the statistical analysis o the results. There are three critical disadvantages when using this approach: convergence o the statistics usually requires a large number o sample trajectories to be propagated,

5 the simulation needs to be repeated or dierent initial distributions, it does not provide the user with analytical inormation, useul or additional analyses. These problems aect both the computational burden associated to the Monte Carlo simulation and its validity or dierent statistics [2]. The previous drawbacks become dramatic when thousands o long-term integrations are required, as or the analysis o possible NEO close encounter with the Earth [16]. DA integration delivers an arbitrary order Taylor expansion o the low o the ODE, which is analytic. Furthermore, the accuracy o the map expansion can be controlled by acting on the expansion order. For these reasons, it is possible to substitute the thousands o pointwise integrations required or classical Monte Carlo simulations with an equal number o map evaluations, i.e. ast polynomials evaluations. The resulting DA-based Monte Carlo simulation can be summarized as: 1. Perorm a single DA integration selecting the expansion order according to the demanded accuracy. 2. Generate random samples based on the statistical distribution o the uncertainty to be propagated. 3. Evaluate the low expansion map or all the samples, requiring only ast polynomial evaluations. 4. Perorm the statistical analysis o the results. The ratio between the computational time o a DA-based Monte Carlo simulation and its pointwise counterpart is given by t n + n s t e n s t, (1) where t n, t e, and t are the computational times o a k-th order DA integration, a low map evaluation, and a pointwise integration, respectively; and n s the number o samples o the Monte Carlo simulation. The computational cost o a Taylor map evaluation, depends on the expansion order, but it is negligible compared to a pointwise integration. For this reason, expression (1) can be approximated by m n s, in which m is the ratio between the computational time o a k-th order DA integration and a pointwise integration (see Fig. 5). The value o m strongly depends on the expansion order, but it is ew orders o magnitude smaller than the number o samples required or a good representation o the statistics. For this reason, the ratio m n s is small, proving that the proposed DA-based Monte Carlo simulation is computationally eicient. As an example, in Sec. 5, Fig. 1 will show that the computational time is reduced by a actor o at least 1 or a typical sample size o 1 virtual asteroids. In case new statistics need to be propagated, it is not necessary to perorm an additional DA integration as only steps 2 4 are required. Furthermore, i the statistical analysis is perormed or a dierent inal time, the possibility o obtaining Taylor expansions with respect to the inal time can be exploited (see Sec. 6.1). Moreover, as the low expansion is analytical, an analytic ramework is delivered. In conclusion, all the major drawbacks o a classical Monte Carlo approach are circumvented. These properties are better highlighted in Sect. 6 by applying the algorithm to the study o Apophis close encounter with the Earth in CED ALGORITHM Let us suppose the close approach o the nominal asteroid occurs at the epoch t, and consider the integration o the asteroid dynamics rom t = t to t = t. Initialize the initial state and the inal integration epoch as DA variables; i.e., [x ] = x + δx [t ] = t + δt, (2) where x is the initial condition corresponding to the nominal asteroid. Using the DA-based RKF78 integrator obtain the map [x ] = x + M x (δx, δt ). (3) The map (3) is the k-th order Taylor expansion o the low with respect to the initial condition

6 and the inal epoch about their nominal values x and t. Based on a mere DA-based computation, the inal solution x can be used to compute the Taylor expansion o distance rom the Earth [d ] = d + M d (δx, δt ). (4) More speciically, map (4) describes how the distance varies depending on the virtual asteroid and the inal integration epoch. Using the derivation operator available in the DA ramework, the Taylor expansion o the derivative d = d(d )/dt can be obtained [d ] = d + M d (δx, δt ). (5) The constant part o the map (5), d, is the derivative o the distance rom the Earth o the nominal solution at its close encounter; i.e., at CED epoch. Consequently, this is a stationary point or the nominal solution, and d =. Then, the map (5) reduces to δd = M d (δx, δt ), (6) in which we omit the bracket operator or the sake o a simpler notation. Consider the map ( ) ( ) ( ) δd Md δx =, (7) δx I x δt which is built by concatenating M d with the identity map or δx. Map (7) can now be inverted to obtain ( ) ( ) 1 ( ) δx Md δd =. (8) δt I x δx This is a ull nonlinear map inversion that is obtained by applying the algorithm illustrated in [5]. This algorithm reduces the map inversion problem to the solution o an equivalent ixed point problem, which can be solved with a ixed amount o eort in the DA setting. Map (4) is then concatenated to the identity map or δt to obtain ( ) ( ) ( ) d Md δx =. (9) δt I t δt Map (9) can now be composed with map (8) to obtain ( ) ( ) ( ) 1 ( ) d Md Md δd =, δt I t I x δx (1) which relates d and δt to the displacements o the derivative o the inal distance δd and o the state vector o the virtual asteroid δx i rom their values. As or the reerence value d =, the necessary condition or CED computation is Substituting into (1) yields ( d δt ) δd =. (11) ( ) ( Md Md = I t I x ) 1 ( δx ). (12) Eventually, map (12) delivers the desired explicit relation between the CED (d ) and the epoch at which it is reached (t + δt ) with the displacement δx i in terms o Taylor polynomials. Given any virtual asteroid belonging to the initial set (which corresponds to a speciic value o the displacement δx i ), the simple evaluation o the polynomials in (12) delivers the CED and the epoch at which it is reached. In this way, the problem highlighted by Milani et al. [15] is solved. 5. CED STATISTICAL ANALYSIS The DA-based Monte Carlo simulation introduced in Sec. 3 is run on map (12) to perorm the nonlinear mapping o the initial uncertainties on the CED. More speciically, 1 virtual asteroids are generated with a normal random distribution with mean value and standard deviation as in Table 1. For each sample, the displacement with respect to the nominal initial conditions is computed and map (12) is evaluated to obtain its CED and the associated epoch. The result is reported in Fig. 6 in terms o probability distribution or the CED. The analysis o the results shows that the mean value is km with a standard deviation o km, thus the possibility o having an Earth impact in 229 is ruled out. For the same virtual asteroids, map (12) is also evaluated to obtain the close encounter epochs. The result is presented in Fig. 7 in terms o the probability distribution o the displacement δt rom the nominal epoch t. The maximum displacement is o the order o 3 s. In Fig. 8 the

7 Probability % CED [km] x 1 4 Figure 6. Monte Carlo analysis o virtual asteroids CED. CED [km] 4.1 x δ t * [s] Figure 8. CED vs δt or 1 virtual asteroids. Probability % Distance rom the Earth [km] 4.1 x CED map evaluation Actual trajectory δ t * [s] Time [MJD2] x 1 4 Figure 7. Monte Carlo analysis o virtual asteroids CED epochs. Figure 9. Accuracy o the CED algorithm: virtual asteroids actual trajectories. 1 virtual asteroids are plotted in the CEDδt plane. For the sake o completeness, an accuracy analysis o the results is presented in Fig. 9. Ten virtual asteroids are randomly selected rom the initial set. For each virtual asteroid, the minimum distance and the corresponding epoch, resulting rom map (12), are reported in the igure. Then, a pointwise integration o the motion o each asteroid is perormed to obtain the proile o Earth s distances shown in the dotted lines. Although the accuracy on the identiication o the epoch o the close encounter is not clearly visualized, due to the very little displacement in t, it is clearly shown that the algorithm is able to accurately identiy the CED values o the resulting trajectories. Figure 1 concludes the analysis by showing the ratio o the computational time between the proposed DA-based Monte Carlo simulation and its pointwise counterpart as a unc- tion o the expansion order when 1 virtual asteroids are considered. It is apparent that the drawback o the higher computational cost required by a DA integration is rewarded by the signiicant time saving achieved by substituting 1 pointwise integrations with the same number o polynomials evaluations. 6. CONCLUSIONS The paper introduced a Monte Carlo simulation based on the high order Taylor expansion o the low o ODE, enabled by the use o dierential algebra. Being based on the replacement o pointwise integrations with ast evaluation o polynomials, the proposed algorithm guarantees signiicant computational time savings. The ac-

8 .2 ACKNOWLEDGMENTS Ratio o computational time % This work was inspired by [2] and was partially conducted under the ESA Ariadna study NEO Encounter 229. The authors are grateul to Dario Izzo or having encouraged the work and to Jon Giorgini or his precious comments Expansion order REFERENCES Figure 1. Percentage o computational time required by a DA-based Monte Carlo run versus a classical Monte Carlo simulation or 1 virtual asteroids. curacy o the algorithm can be suitably tuned by varying the low expansion order. Furthermore, the availability o analytic Taylor expansions and the use o DA embedded tools as map inversion, composition, and derivation allows the user to compute arbitrary order maps o the quantities on which the statistical analysis is perormed; thus, the algorithm is not limited to the low o ODE. More speciically, a technique or the automatic computation o both CED and CED epochs or all the virtual asteroids belonging to the initial uncertainty cloud has been developed. The eiciency and eectiveness o the methods are proven by applying them to the analysis o Apophis close encounter with the Earth occurring in April 229. In particular, it is shown that the nonlinear mapping o uncertainties can be perormed or any complex and arbitrary dynamics, even when long-term integrations are required; a ith order expansion increases the accuracy o the computation o the CED by approximately two orders o magnitude with respect to classical linear methods; the expansion in time allows or the proper identiication o the CED epoch or all the virtual asteroids. As an additional result, the occurrence o an impact with the Earth in April 229 can be ruled out. [1] Battin, R.H.: An Introduction to the Mathematics and Methods o Astrodynamics. AIAA Education Series, New York, [2] Berz, M., Joh, K., Nolen, J.A., Sherrill, B.M., and Zeller, A.F.: Reconstructive correction o aberration in nuclear particle spectrographs. Physical Review C, 47, pp , (1993). [3] Bernelli-Zazzera, F., Berz, M., Lavagna, M., Makino, K., Armellin, R., Di Lizia, P., Jagasia, R., and Topputo, F.: NEO Encounter 229: Orbital Prediction via Dierential Algebra and Taylor Models. Final Report, Ariadna id: 8-433, Contract No. 2271/6/NL/HI (29). [4] Berz, M.: Dierential Algebraic Techniques. Entry in Handbook o Accelerator Physics and Engineering, World Scientiic, New York, (1999a). [5] Berz, M.: Modern Map Methods in Particle Beam Physics. Academic Press, (1999b). [6] Berz, M., and Makino, K.: COSY INFIN- ITY version 9 reerence manual. MSU Report MSUHEP-683, Michigan State University, East Lansing, MI 48824, pp 1-84, (26). [7] Crassidis, J.L., and Junkins, J.L.: Optimal Estimation o Dynamics Systems. CRC Press LLC, Boca Raton, FL, pp , (24). [8] Chesley, S.R., and Milani, A.: An automatic Earth-asteroid collision monitoring system. Bull. Am. Astron. Soc., 32, pp 682, (2). [9] Chodas, P.W., and Yeomans, D.K.: Predicting close approaches and estimating impact probabilities or near-earth projects. AAS/AIAA Astrodynamics Specialists Conerence, Girdwood, Alaska, (1999).

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