An Approach for Nonlinear Uncertainty Propagation: Application to Orbital Mechanics Daniel Giza∗ , Puneet Singla† , Moriba Jah‡ An approach for nonlinear propagation of orbit uncertainties is discussed while making use of the Fokker-Planck-Kolmogorov Equation (FPKE). The central idea is to replace evolution of initial conditions for a dynamical system with the evolution of a probability density function (pdf ) for state variables. The transition pdf corresponding to dynamical system state vector is approximated by using a finite Gaussian mixture model. The mean and covariance of different components of the Gaussian mixture model are propagated through the use of an Unscented Kalman Filter (UKF). Furthermore, the unknown amplitudes corresponding to different components of the Gaussian mixture model are found by minimizing the FPKE error over the entire volume of interest. This leads to a convex quadratic minimization problem guaranteed to have a unique solution. The two-body problem model with non-conservative atmospheric drag forces and initial uncertainty will be used to show the efficacy of the ideas developed in this paper.
I.
Introduction
In July, 1994, the comet Shoemaker-Levy 9 collided with Jupiter at over 133,000 mph. In March, 2004, a group of about large 30 meteors passed by the Earth at only one tenth the distance to the moon; these space rocks were discovered only three days prior to the event. A spectacular fireball lit up the sky above Northern Sudan on October 7, 2008. This explosion was caused by the atmospheric entry of a small near-Earth asteroid, estimated to ∗ Graduate Student, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-14260, Email:
[email protected]. † Assistant Professor, Senior AIAA, AAS Member, Department of Mechanical & Aerospace Engineering, University at Buffalo, Buffalo, NY-14260, Email:
[email protected]. ‡ Director, Advanced Sciences and Technology Research Institute for Astrodynamics (ASTRIA), Air Force Research Laboratory, 535 Lipoa Parkway, Suite 200, Kihei, HI-96753
1 of 19 American Institute of Aeronautics and Astronautics
be no more than a few meters in diameter. Asteroid 2004 MN4 is predicted to make a very close encounter in April of 2029, followed by other approaches in 2034 and 2036. The uncertainty of the 2029 approach (± 1 Earth radii) is quite large compared to the predicted miss distance of 5.6 Earth radii. This close encounter is not presently considered a high risk for collision, however the subsequent encounters have larger uncertainties and cannot be accurately predicted due to the nonlinearity of the dynamics and the large uncertainty associated with the 2029 approach. According to a NASA catalog, there are a total of 1015 potentially hazardous asteroids. After cataloging, the next challenge toward ensuring our safety is to determine the path of these near Earth objects (NEOs) for threat analysis. Similarly, the increasing amount of debris and inactive resident space objects (RSOs), particularly those with high area-to-mass ratios, in both the low-earth-orbit (LEO) and geosynchronous-earthorbit (GEO) regime pose a threat to active RSOs and must be accurately tracked for threat analysis. A mechanism to represent the uncertainty is necessary before analysis can be done in an efficient and consistent manner. Probabilistic means of representing uncertainty has been explored extensively and provides the greatest wealth of knowledge. The most common method for representing orbital uncertainty is to approximate the initial distribution using a Gaussian model derived from the orbit determination, and use linear error theory to propagate the mean and covariance of the Gaussian model forward in time. This can lead to significant errors when propagating uncertain orbits for large amounts of times.1–4 In Refs.,5, 6 judicious choices for coordinate systems is advocated to decrease the uncertainty propagation errors while making use of linear error theory. In addition to this, several approximate techniques exist in the literature to approximate the initial condition uncertainty evolution, the most popular being Monte Carlo (MC) methods,7 Gaussian closure,8 Equivalent Linearization,9 and Stochastic Averaging.10, 11 All of these algorithms except Monte Carlo methods are similar in several respects, and are suitable only for linear or moderately nonlinear systems, because the effect of neglected higher order terms can lead to significant errors. Monte Carlo methods require extensive computational resources, and become increasingly infeasible for 2 of 19 American Institute of Aeronautics and Astronautics
high-dimensional dynamic systems.12 Furthermore, all these approaches provide only an approximate description of the uncertainty propagation problem by restricting the solution to a small number of parameters - for instance, the first N moments of the sought pdf. For stochastic continuous dynamic systems the exact evolution of state pdf is given by the Fokker-Planck-Kolmogorov Equation (FPKE).13 Park et al.14 has discussed the use of FPKE to analyze the spacecraft trajectory statistics by incorporating higher order Taylor series term in the spacecraft dynamics. Analytical solutions for the Fokker-Planck Kolmogorov equation exist only for a stationary probability density function and are restricted to a limited class of dynamical systems.13, 15 In this paper, we explore the use of recently developed Adaptive Gaussian Mixture Model approach16, 17 for accurate orbit uncertainty propagation while incorporating the solution to the FPKE.
II.
Equations of Motion
The classical differential equations governing the perturbed two-body problem1 are given by x¨ +
µx r3
= adx (t, x, y, z, x, ˙ y, ˙ z), ˙ r 2 = x2 + y 2 + z 2
y¨ +
µy r3
= ady (t, x, y, z, x, ˙ y, ˙ z), ˙ (˙) ≡ d()/dt
z¨ +
µz r3
= adz (t, x, y, z, x, ˙ y, ˙ z), ˙ µ ≡ G(m1 + m2 )
(1)
or in compact form: ¨r +
µr = ad r3
(2)
Given the initial conditions{r(t0 ), r˙ (t0 )}, and prescription of the perturbing acceleration function ad , Eqs. (1), (2) can be solved for the Cartesian coordinate trajectory. However, the exact initial conditions and perturbing acceleration profiles are typically not known. Throughout most of the modern history of physics, it has been assumed that it is possible to shrink the uncertainty in the final dynamical prediction by measuring the initial conditions to greater and greater accuracy. But Poincar´e noticed that certain astronomical systems did not seem to obey the rule that increasing the accuracy of the initial conditions would effect the 3 of 19 American Institute of Aeronautics and Astronautics
final prediction in a corresponding way.18 Poincar´e showed that a very tiny imprecision in the initial conditions would grow in time at an enormous rate. Thus, two nearly-indistinguishable sets of initial conditions for the same system would result in two final predictions that differ vastly from each other. Similar observations were made by Lorenz in 1963 while studying a primitive model of atmospheric current. He showed that even the smallest imaginable discrepancy between two sets of initial conditions would always result in a large discrepancy at later or earlier times.19, 20 Thus, the presence of chaotic systems in nature seems to place a limit on our ability to apply deterministic physical laws to predict motions with any degree of certainty. This means that in order to make long-term forecasts with any degree of accuracy, it is necessary to understand how the initial conditions and model uncertainty propagate through the nonlinear mathematical models.
III.
Uncertainty Propagation
In conventional deterministic systems, the system state assumes a fixed value at any given instant of time. However, in stochastic dynamics it is a random variable and its time evolution is given by a stochastic differential or difference equation:
˙ x(t) = f (t, x) + g(t, x)Γ(t)
(3)
where, g(t, x)Γ(t) is a stochastic term representing the combined effect of un-modeled dynamics, and external disturbances acting on the system. In the orbit uncertainty propagation problem, this terms can constitute un-modeled external disturbances such as solar radiation pressure, atmosphere drag for low-earth-satellite orbit etc. We further assume Γ(t) to be a zero mean white-noise process with the correlation matrix Q. The uncertainty associated with the state vector x(t) is usually characterized by a time dependent state pdf p(t, x). In essence, the study of stochastic systems reduces to finding the nature of the time-evolution of the system-state pdf. For stochastic continuous-time dynamic systems, the exact evolution
4 of 19 American Institute of Aeronautics and Astronautics
of state pdf is given by the Fokker-Planck-Kolmogorov equation (FPKE)13 ∂ ∂p(t, x) 1 ∂ 2p T T p(t, x) = f (x, t) + T r g(t, x(t))Qg (t, x(t)) ∂t ∂x 2 ∂x∂xT
(4)
The FPKE is a formidable equation to solve, because of the following issues: 1. Positivity of the pdf: p(t, x)dx ≥ 0. 2. Normalization constraint of the pdf:
R Rn
p(t, x)dx = 1.
3. No fixed Solution Domain: how to impose boundary conditions in a finite region and restrict numerical computations to the volume of interest. Analytical solutions for the FPKE exist only for a stationary pdf and are restricted to a limited class of dynamical systems.13, 15 Thus researchers are actively looking at numerical approximations to solve the FPKE,21–26 generally using the variational formulation of the problem. However, these methods suffer from the “curse of dimensionality” since one needs to discretize the space over which the pdf mass lies and this discretization comes at an expense of overwhelmingly cost of computation.26 In this paper, we will make use of the recently developed adaptive Gaussian mixture model approach to solve the FPKE efficiently. The main idea of this approach is to approximate the joint probability density function by a weighted sum of independent Gaussian pdfs. The weights associated with each Gaussian elements are selected so that the FPKE residual error is minimized over an area of interest.16, 17, 27 An advantage of this approach is that the problem of solving FPKE is posed as a convex optimization problem with a unique solution. In Refs. [16,17,27–29], the effectiveness of this approach has been demonstrated in solving the FPKE for higher dimension problems including the 6-D spacecraft attitude estimation problem.29
IV.
Gaussian Mixture Model
The main idea behind the Adaptive Gaussian Mixture model approach, is to solve the FPKE by approximating the forecast pdf using a finite sum of Gaussian components and 5 of 19 American Institute of Aeronautics and Astronautics
tools from convex optimization. Here, we briefly discuss the main idea and details can be found in Refs. [16, 17, 27]. Let us consider the following equation depicting the Gaussian mixture model approximation for the forecast density function, p(t, x)
pˆ(t, x) =
N X
w i pg i
where pgi = N (x(t) | µi , Pi )
(5)
i=1
where µi and Pi represent the mean and covariance of the ith component of the Gaussian mixture, N (x(t) | µi , Pi ), respectively and wi denotes the amplitude of ith Gaussian in the mixture. The positivity and normalization constraint on the mixture pdf, pˆ(t, x), leads to following constraints on the amplitude vector: N X
wi = 1, wi ≥ 0, ∀i
(6)
i=1
In Ref. [30], it is shown that since all the components of the mixture pdf of Eq. (5) are Gaussian and thus, only estimates of their mean and covariance need to be maintained in order to obtain the optimal state estimates. These can be propagated by using a continuoustime derivation of the Unscented Kalman Filter,31 which uses a set of deterministically chosen sigma-points that capture the mean and covariance of the initial distribution. The propagation equations for mean and covariance are given as
Xi = [µi . . . µi ] +
√ c [0 A −A] , c = α2 (n + κ)
(7)
µ˙ i = f (Xi )wm
(8)
˙ i = Xi Wf T (Xi ) + f (Xi )WXi T + g(t, µi )QgT (t, µi ) P
(9)
where Xi is the n × 2n + 1 matrix of sigma-points and the weight vector, wm , and weight
6 of 19 American Institute of Aeronautics and Astronautics
matrix, W , are given by
λ = α2 (n + κ) − n (mean)
W0
=
λ n+λ
λ (n + λ) + (1 − α2 + β) 1 (mean) , j = 1, . . . , 2n Wj = (2(n + λ)) 1 (cov) Wj = , j = 1, . . . , 2n (2(n + λ)) h iT (mean) (mean) wm = W0 . . . W2n (cov)
W0
=
(10)
(cov) (cov) . . . W2n × (I − [wm . . . wm ])T W = (I − [wm . . . wm ]) × diag W0
(11)
The constants α, β, and κ in the above equations are constant parameters of the method. The spread of sigma points is determined by α and is typically a small positive value, i.e. 1×10−4 ≤ α ≤ 1. The parameter β is used to incorporate prior knowledge of the distribution, and for is optimally chosen as 2 for a normal distribution. The parameter κ can be used to exploit knowledge of the distributions higher moments, and for higher order systems choosing κ = 3 − n minimizes the mean-squared-error up to the fourth order.2 Notice that the weights wi corresponding to each Gaussian component are also unknown. Hence, the idea is to use FPKE error as a feedback to update the amplitude of different Gaussian components in the mixture pdf. In other words, we seek to minimize the FPKE error under the assumption of Eqs. (5), (6), (8) and (9). Substituting Eq. (5) in Eq. (4) leads to e(t, x) =
∂ pˆ(t, x) − LF P (ˆ p(t, x)) ∂t
(12)
where we denote LF P (ˆ p(t, x)) as the Fokker-Planck operator. The application of this operator
7 of 19 American Institute of Aeronautics and Astronautics
is given by,
LF P (ˆ p(t, x)) =
N X
wit LF P (pgi )
i=1
= lFP T w
(13)
where w is the vector of weights at time t and the elements of lFP are given by the application of the Fokker-Planck operator to the individual Gaussian components: ∂pTgi ∂f (t, x) 1 ∂ 2 pg i LF P (pgi ) = − f (t, x) − pgi Tr + Tr Q ∂x ∂x 2 ∂x∂xT
(14)
The first term in Eq.(12) is the time derivative of the probability density function approximated by the Gaussian mixture, and is given by, ! N T ∂p ∂ pˆ(t, x) X ∂p gi ˙ g = w˙ it pgi + wit ii µ˙ i + wit Tr Pi ∂t ∂µ ∂P i i=1
(15)
˙ are given by Eq.(8)-(9), and the derivative The derivatives of the first two moments, µ˙ and P, of the weights, w˙ i , is obtained by the following finite difference approach: w˙ it =
1 0 (wit − wit ) where t0 = t + ∆t ∆t
(16)
Substituting Eq.(16) into Eq.(15), we obtain N N X X ∂ pˆ(t, x) 1 0 = pgi wit + ∂t ∆t i=1 i=1
! ∂pTgi ∂pgi ˙ 1 µ˙ i + Tr Pi − pg wit ∂µi ∂Pi ∆t i | {z }
(17)
mDT i
=
1 0 pg T wt + mDT T wt ∆t
(18)
0
where wt is the vector of new weights to be found, pg is the vector of Gaussian mixture components, and the elements of mDT are shown in the above equation. Furthermore, the different derivatives of the Gaussian components in the above equations are computed using 8 of 19 American Institute of Aeronautics and Astronautics
these analytical expressions, ∂pgi ∂µi ∂pgi ∂Pi ∂pgi ∂x 2 ∂ pg i ∂x∂xT
= P−1 i (x − µi ) pgi i 1 h −1 T −1 −1 = Pi (x − µi ) (x − µi ) Pi − Pi pgi 2 = −P−1 i (x − µi ) pgi h i −1 = (Pi )−1 (x − µi ) (x − µi )T P−1 pg i − P i i
Substituting Eq.(18) and Eq.(13) into the FPKE error equation, Eq.(12), gives
e(t, x) =
1 pg T wt0 + (mDT − lFP )T wt ∆t
(19)
Now, at a given time instant, after propagating the mean, µi , and the covariance, Pi , of individual Gaussian elements using Eqs. (8) and (9), we seek to update weights by minimizing the FPKE error over some volume of interest V . 1 min 0 2 wit s.t
Z
V N X
e2 (t, x)dx
(20)
0
wit = 1
i=1 0
wit ≥ 0,
i = 1, · · · , N 0
Because the FPKE error, Eq.(19), is linear in the new Gaussian component weights, wit , the problem can be written as a quadratic programming problem.
min 0 wit
1 T w 0 Mc wt0 + wtT0 Nc wt 2 t
s.t 1TN ×1 wt0 = 1 wt0 ≥ 0N ×1
9 of 19 American Institute of Aeronautics and Astronautics
(21)
Where 1N ×1 ∈ RN ×1 is a vector of ones, 0N ×1 ∈ RN ×1 is a vector of zeros and the matrices Mc ∈ RN ×N and Nc ∈ RN ×N are given by
Mc Nc
Z 1 = pg pg T dx ∆t2 ZV 1 pg (mDT − lFP )T dx = ∆t
(22) (23)
V
The individual components of these two matrices are given by:
mcij
1 1 −1/2 = 2 |2π(Pi + Pj )| exp − (µi − µj )T ∆t 2 ×(Pi + Pj )−1 (µi − µj ) for i 6= j 1 |4πPi |−1/2 ∆t2 Z T ∂pgj ∂pgj ˙ 1 1 µ˙ j + Tr pg Pj − = pg i ∆t ∂µj ∂Pj ∆t j V ∂pTgj ∂ 2 pgj ∂f (t, x) 1 + f (t, x) + pgj Tr − Tr Q dx ∂x ∂x 2 ∂x∂xT
mcii = ncij
(24) (25)
(26)
A major challenge in solving the convex minimization problem of Eq. (21) is the need to evaluate integrals involving Gaussian pdfs over volume V . This volume V can be defined by making use of the fact that the mass of a Gaussian pdf is concentrated in a finite volume about its mean. This is one of the major advantages of using the Gaussian mixture model approximation, as it automatically defines the space over which probability mass lies. Also, these integrals can be computed exactly for polynomial nonlinearity and in general can be approximated by using Gaussian Quadrature, Monte Carlo integration or Unscented Transformation methods.32 While in lower dimensions the Unscented Transformation and Gaussian Quadrature methods are mostly equivalent, the Unscented Transformation is computationally more appealing in evaluating integrals for higher dimensions as the number of points taken to compute the integral grows only linearly with the number of dimensions.
10 of 19 American Institute of Aeronautics and Astronautics
V.
Low Earth Orbit with Atmospheric Drag
Non-conservative forces in orbital mechanics very quickly lead to the evolution of nonGaussian state-pdfs, even when the initial distribution is Gaussian. This is especially true when the object of interest has a high area-to-mass ratio (HAMR). These HAMR objects were first discovered in the geosynchronous regime and are hypothesized to be part of the rapidly increasing population of space debris.33 After the recent collision of a US Iridium satellite and the defunct Russian Cosmos 2251 satellite, the debris had to be accurately tracked in order to analyze threat to other satellites in similar orbits. Similarly, some initial speculation about HAMR objects indicates that several may be multi-layered insulation material that have become separated from its spacecraft of origin by collision or some other means. In the geosynchronous regime HAMRs are largely effected by solar radiation pressure as well as third-body effects. In low-earth orbit the effect of atmospheric drag dominates, and this is the problem focused on below. Consider the dynamics of an object in low-earth-orbit (LEO) that is affected by nonconservative atmospheric drag forces, whose planar equations of motion are given by34 µx ˙ y), ˙ = aDx (t, x, y, x, r3 µy ˙ y), ˙ y¨ + 3 = aDy (t, x, y, x, r
x¨ +
aD =
1 Cd A 2 vrel ρvrel 2 m |vrel |
ρ = ρ0 e
−
(r−RL ) h
where Cd is the coefficient of drag, A is the cross-sectional area, m is the mass of the object, and ρ is the atmospheric density at a given altitude. The atmospheric density model is assumed to be an exponential model with reference density ρ0 . It is also worth noting that the vrel is not the velocity state vector, but rather the velocity relative to Earth’s atmosphere. For simulation, we assume that we have an initial state-pdf that is Gaussian with mean
11 of 19 American Institute of Aeronautics and Astronautics
and covariance shown below, 6 6.6032 × 10 0 µ0 = 0 3 7.7695 × 10
6
1.78 × 10 0 P0 = 0 0
0 2.50 × 105 0 0
0
0 0 0 6.25 0 0 25
which correspond to a starting altitude of 225 km. The covariance matrix reflects a larger uncertainty in the radial position than in the in-track position as well as a larger uncertainty in the tangential velocity than in radial velocity. The ballistic coefficient is chosen as B = 1.4, where B =
Cd A , m
which is consistent with a HAMR object. Other assumptions include a
circular orbit and the absence of process noise under the assumption that the perturbing atmospheric drag forces are being accurately modeled. A Monte Carlo simulation is run in which 50,000 samples are taken from the initial distribution and propagated for 2 orbit periods. The resulting histograms are taken to be the true conditional pdf at a given time to which the Adaptive Gaussian Mixture results can be compared. Several other mixture components are chosen as the sigma points from −3σ to 3σ of the initial distribution along the radial position and the tangential velocity, in this case x and y˙ respectively, as those are the two states with the highest uncertainty and have the greatest effect on the final state-pdf. This leads to a total of 50 mixture components, as all combinations of the sigma points for these two states are taken. The initial Gaussian distribution is assigned a weight of 1, and the mixture components placed along the sigma points are assigned a weight of 0. In Fig.(1), the histograms created from the Monte Carlo simulation and the Adaptive Gaussian Mixture pdfs are shown. It is easily seen that the atmospheric drag is beginning to curve and skew the distribution and that the Adaptive Gaussian mixture accurately captures this behavior. In fact, after 1 orbital period the initial distribution is assigned a weight of 0, evidencing that the initial Gaussian distribution no longer accurately captures the behavior of the state-pdf. The time-evolution of the weights
12 of 19 American Institute of Aeronautics and Astronautics
for the initial distribution and the dominant Gaussian components after two orbital periods, T , are shown in Table 1. The coherence of the Adaptive Gaussian Mixture results to the Monte Carlo results is best seen from the conditional pdf contours. In Fig.(2), the contours of the Monte Carlo simulation, Adaptive Gaussian Mixture pdfs, as well as the Unscented Kalman Filter propagation of the initial Gaussian distribution are shown. These contours are generated at 1% of the pdf’s peak value done because although the contours of constant probability are easily generated for the Unscented Kalman Filter, the Adaptive Gaussian mixture and the Monte Carlo samples do not have an equivalent 3σ contour, and plotting the 1% contour is done for consistency. It is clear that the Unscented Kalman Filter and the Adaptive Gaussian mixture pdf are initially identical and remain similar for some time. Eventually, however, the Unscented Kalman Filter no longer accurately represents the area of uncertainty given by the Monte Carlo samples contour while the Adaptive Gaussian Mixture contour is able to very accurately capture the non-Gaussian behavior.
VI.
Conclusions
Orbit uncertainty propagation is an important problem to study while designing the orbit for man-launched spacecraft or doing the risk-analysis for NEO collision with Earth. This paper presents the formulation for accurate uncertainty propagation through orbital mechanics while making use of the Fokker-Planck-Kolmogorov Equation (FPKE). A Gaussian mixture model is used to approximate the state transition pdf and furthermore, a convex optimization problem is posed to update unknown parameters of the mixture model. This approach is shown to provide much more accurate uncertainty propagation than traditional methods that simply maintain a mean and covariance of an initial distribution approximated by a single Gaussian, and is very attractive for orbital problems involving high area-to-mass ratio objects under the non-conservative perturbing forces of atmospheric drag and solar radiation pressure.
13 of 19 American Institute of Aeronautics and Astronautics
(a) MC Histogram - 0.5 Orbits
(b) AGM PDF - 0.5 Orbits
(c) MC Histogram - 1 Orbit
(d) AGM PDF - 1 Orbit
(e) MC Histogram - 1.5 Orbits
(f) AGM PDF - 1.5 Orbits
(g) MC Histogram - 2 Orbits
(h) AGM PDF - 2 Orbits
Figure 1. Conditional PDFs
14 of 19 American Institute of Aeronautics and Astronautics
(a) Initial PDF Contours
(b) 0.25 Orbits
(c) 0.5 Orbits
(d) 0.75 Orbits
(e) 1 Orbit
(f) 1.25 Orbits
(g) 1.5 Orbits
(h) 1.75 Orbits
(i) 2 Orbits
Figure 2. Conditional PDF Contours
15 of 19 American Institute of Aeronautics and Astronautics
Table 1. Time-Evolution of Dominant Gaussian Components Weights
Time 0 0.1T 0.2T 0.3T 0.4T 0.5T 0.6T 0.7T 0.8T 0.9T 1.0T 1.1T 1.2T 1.3T 1.4T 1.5T 1.6T 1.7T 1.8T 1.9T 2.0T
w1 1.0000 0.9995 0.9961 0.9654 0.3349 0.0126 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
w19 0.0000 0.0000 0.0005 0.0011 0.0593 0.1003 0.1022 0.1089 0.1156 0.1202 0.1220 0.1210 0.1177 0.1135 0.1105 0.1129 0.1230 0.1393 0.1560 0.1674 0.1712
w20 0.0000 0.0000 0.0003 0.0005 0.0352 0.0570 0.0581 0.0612 0.0648 0.0673 0.0682 0.0676 0.0657 0.0632 0.0612 0.0620 0.0669 0.0754 0.0850 0.0924 0.0953
w26 0.0000 0.0000 0.0006 0.0060 0.1230 0.1877 0.1952 0.1999 0.2022 0.2030 0.2031 0.2034 0.2039 0.2047 0.2055 0.2068 0.2082 0.2072 0.2025 0.1961 0.1928
w27 0.0000 0.0000 0.0004 0.0046 0.0697 0.1063 0.1117 0.1140 0.1150 0.1153 0.1153 0.1154 0.1156 0.1159 0.1162 0.1167 0.1172 0.1167 0.1148 0.1126 0.1114
16 of 19 American Institute of Aeronautics and Astronautics
References 1
R. H. Battin. An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education
Series. AIAA, Reston, VA, 1999. 2
J. L. Crassidis and J. L. Junkins. Optimal Estimation of Dynamical Systems. CRC Press, Boca Raton,
FL, 2004. 3
P. W. Chodas and D. K. Yeomans. Orbit determination and estimation of impact probability for near
earth objects. In Proceedings of the Guidance and Control, volume 101 of Advances in the Astronautical Sciences, pages 21–40, 1999. 4
J. Devis, P. Singla, and J. L. Junkins. Identifying near-term missions and impact keyholes for asteroid
99942 apophis. 7th International Conference On Dynamics and Control of Systems and Structures in Space. London, England, July 2006. 5
J. L. JUNKINS and P. SINGLA. How nonlinear is it? a tutorial on nonlinearity of orbit and attitude
dynamics. Journal of Astronautical Sciences, 52(1-2):7–60, 2004. keynote paper. 6
J. L. Junkins, M. Akella, and K. Alfriend. Non-gaussian error propagation in orbit propagation. Jornal
of the Astronaumical Sciences, 44(4), 1996. 7
A. Doucet, N. de Freitas, and N. Gordon. Sequential Monte-Carlo Methods in Practice. Springer-
Verlag, April 2001, 6-14. 8
R. N. Iyengar and P. K. Dash. Study of the random vibration of nonlinear systems by the gaussian
closure technique. Journal of Applied Mechanics, 45:393–399, 1978. 9
J. B. Roberts and P. D. Spanos. Random Vibration and Statistical Linearization. Wiley, 1990, 122-176.
10
T. Lefebvre, H. Bruyninckx, and J. De Schutter. Kalman filters of non-linear systems: A comparison
of performance. International journal of Control, 77(7):639–653, 2004. 11
T. Lefebvre, H. Bruyninckx, and J. De Schutter. Comment on a new method for the nonlinear
transformations of means and covariances in filters and estimators. IEEE Transactions on Automatic Control, 47(8), 2002. 12
F. Daum and J. Huang. Curse of dimensionality and particle filters. Aerospace Conference, 2003.
Proceedings. 2003 IEEE, 4:1979–1993, March 8-15, 2003. 13
H. Risken. The Fokker-Planck Equation: Methods of Solution and Applications. Springer, 1989, 1-12,
32-62. 14
R. S. Park and D. J. Scheeres. Nonlinear mapping of gaussian statistics: Theory and application to
spacecraft design. AIAA Journal for Guidance, Control and Dynamics, 29(6), Nov.-Dec. 2006.
17 of 19 American Institute of Aeronautics and Astronautics
15
A. T. Fuller. Analysis of nonlinear stochastic systems by means of the fokker-planck equation. Inter-
national Journal of Control, 9:6, 1969. 16
G. Terejanu, P. Singla, T. Singh, and P. D. Scott. Uncertainty propagation for nonlinear dynamical
systems using gaussian mixture models. In Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, HI, 2008. 17
Gabriel Terejanu, Puneet Singla, Tarunraj Singh, and Peter D. Scott. Uncertainty propagation for
nonlinear dynamical systems using gaussian mixture models. Journal of Guidance, Control, and Dynamics, In Press, 2008. 18
H. Poincar´e. Science and Hypothesis. Dover Publications, 1952.
19
E. N. Lorenz. Deterministic nonperiodic flow. Journal of Atmospheric Sciences, 20(130), 1963.
20
E. N. Lorenz. The Essence of Chaos. Univ. of Washington Press, 1993.
21
H. C. Lambert, F. E. Daum, and J. L. Weatherwax. A split-step solution of the fokker-planck equation
for the conditional density. Signals, Systems and Computers, 2006. ACSSC ’06. Fortieth Asilomar Conference on, pages 2014–2018, Oct.-Nov. 2006. 22
M. Kumar, P. Singla, S. Chakravorty, and J. L. Junkins. A multi-resolution approach for steady state
uncertainty determination in nonlinear dynamical systems. In 38th Southeastern Symposium on System Theory, 2006. 23
M. Kumar, P. Singla, S. Chakravorty, and J. L. Junkins. The partition of unity finite element approach
to the stationary fokker-planck equation. In 2006 AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone, CO, Aug. 21-24, 2006. 24
G. Muscolino, G. Ricciardi, and M. Vasta. Stationary and non-stationary probability density function
for non-linear oscillators. International Journal of Non-Linear Mechanics, 32:1051–1064, 1997. 25
M. D. Paola and A. Sofi. Approximate solution of the fokker-planck-kolmogorov equation. Probabilistic
Engineering Mechanics, 17:369–384, 2002. 26
P. Singla and J. L. Junkins. Multi-Resolution Methods for Modeling and Control of Dynamical Systems.
Applied Mathematics and Nonlinear Science. Chapman & Hall/CRC, Boca Raton, FL, July 2008. 27
P. Singla and T. Singh. A gaussian function network for uncertainty propagation through nonlinear
dynamical system. 18th AAS/AIAA Spaceflight Mechanics Meeting, Galveston, TX, Jan. 27-31, 2008. 28
Gabriel Terejanu, Puneet Singla, Tarunraj Singh, and Peter D. Scott. A novel gaussian sum filter
method for accurate solution to nonlinear filtering problem. 2008 International Conference on Information Fusion.
18 of 19 American Institute of Aeronautics and Astronautics
29
G. Terejanu, J. George, and P. Singla. An adaptive gaussian sum filter for the spacecraft attitude
estimation problem. Accepted for publication in the Journal of Astronautical Sciences, May, 2009. 30
H. Alspach, D.; Sorenson. Nonlinear bayesian estimation using gaussian sum approximations. Auto-
matic Control, IEEE Transactions on, 17(4):439–448, 1972. 31
S. Sarkka. On unscented kalman filtering for state estimation of continuous-time nonlinear systems.
Automatic Control, IEEE Transactions on, 52(9):1631–1641, 2007. 32
A. Honkela. Approximating nonlinear transformations of probability distributions for nonlinear inde-
pendent component analysis. In IEEE International Joint Conference on Neural Networks, 2004. 33
T. Flohrer T. Schildknecht, R. Musci. Properties of the high area-to-mass ratio space debris population
at high altitudes. Advances in Space Research, 41:1039–1045, 2008. 34
David A. Vallado. Fundamentals of Astrodynamics and Applications. Microcosm, 2007.
19 of 19 American Institute of Aeronautics and Astronautics
