Analytical Non-Linear Uncertainty Propagation: Theory And Implementation

Transcription

1 Analytical Non-Linear Uncertainty Propagation: Theory And Implementation Kohei Fujimoto 1), Daniel J. Scheeres 2), and K. Terry Alfriend 1) May 13, ) Texas A&M University 2) The University of Colorado at Boulder

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4 Introduction Consistent propagation of uncertainty One topic of recent interest in Space Situational Awareness (SSA) and Orbit Determination (OD) is the consistent representation of an observed object s uncertainty August 7, rd US-China Tech. Interchange 4

5 Introduction The proposed technique Special solution to the Fokker-Planck eqn. for deterministic systems Non-conservative forces State transition tensor (STT) King-Hele s analytic theory of motion under atm. drag Analytic expression of the uncertainty for all time 8/7/2013 AMOS

6 Theory The proposed technique Analytically expressed pdfs can be incorporated in many practical tasks in space situational awareness (SSA) Rapid propagation of the mean and covariance of the uncertainty for arbitrary initial pdfs Does not involve any sampling Accuracy determined a priori for two-body approximation Model parameter uncertainty readily implemented Addition of ballistic coefficient and its corresponding uncertainty in the state vector Use in non-linear Bayesian estimator for orbit determination and conjunction assessment August 7, rd US-China Tech. Interchange 6

7 Theory Special soln. to Fokker-Planck eqn. For a system that satisfies the Itô stochastic differential equation, the time evolution of a pdf p over X at time t is described as: As a special solution for deterministic systems, the probability is an integral invariant: August 7, rd US-China Tech. Interchange 7

8 Theory Special soln. to Fokker-Planck eqn. Consequently, analytical expr. for the initial pdf + aanalytical expr. for the solution flow = analytical expr. for the pdf for all time Propagation is only a matter of changing the value for time August 7, rd US-China Tech. Interchange 8

9 Theory Special soln. to Fokker-Planck eqn. The X/ X 0-1 term accounts for the change in phase volume due to non-conservative forces. For Hamiltonian systems, X/ X 0-1 = 1 from Liouville s theorem, and therefore: analytical expr. for the initial pdf + aanalytical expr. for the solution flow = analytical expr. for the pdf for all time Propagation is only a matter of changing the value for time August 7, rd US-China Tech. Interchange 9

10 Theory State transition tensors (STTs) Some dynamical systems do not have a closed-form solution, but we can nevertheless express the solution analytically in approximate form via the State Transition Tensor (STT) concept. Take a Taylor series expansion of ϕ(t; X 0, t 0 ) about a reference trajectory X * (x = X X * ) [Park and Scheeres, 2006]: STTs are a generalization of the state transition matrix to any arbitrary order: August 7, rd US-China Tech. Interchange 10

11 Theory State transition tensors (STTs) If ϕ(t; X 0, t 0 ) known analytically, Otherwise, solve differential equation: where A is the local dynamics tensor: and ICs: August 7, rd US-China Tech. Interchange 11

12 Theory State transition tensors (STTs) Can also analytically propagate the mean m(t) and variance-covariance matrix [P] of the pdf: Deterministic August 7, rd US-China Tech. Interchange 12

13 Theory State transition tensors (STTs) Can also analytically propagate the mean m(t) and variance-covariance matrix [P] of the pdf: Hamiltonian where E[ ] is the expected value operator August 7, rd US-China Tech. Interchange 13

14 Implementation Simulation setup (I) Example I Propagation of 3-σ ellipsoid for object in circular LEO influenced by atmospheric drag a Initial 300 km altitude, 15 deg inclination Propagation over days for two epochs reflecting a difference in geomagnetic activity Calm (Feb. 8, 2009) Stormy (Jul. 11, 2000) August 7, rd US-China Tech. Interchange 14

15 Implementation Simulation setup (I) Monte Carlo a Numerical integration of the dynamics aa 2000 sample points distributed normally: σ a = 20 km, σ M = 0.01 deg a Ballistic coefficient allowed to fluctuate sinusoidally a Density modeled with NRLMSISE-00 August 7, rd US-China Tech. Interchange 15

16 Implementation Simulation setup (I) Analytic 1000 points taken on the 3-σ ellipsoid of the initial Gaussian distribution σ a = 20 km, σ M = 0.01 deg, σ BC = 5 kg/m 2 a Analytic solution of King-Hele expanded up to 4 th order assuming that the atmosphere is spherically symmetrical density model is exponential with constant parameters in time rotates at the same angular rate as the Earth August 7, rd US-China Tech. Interchange 16

17 Implementation Applying King-Hele s results For an initially circular orbit, only the semi-major axis, and consequently the mean anomaly, change with time The non-zero terms in the n-th order STT are August 7, rd US-China Tech. Interchange 17

18 Implementation Applying King-Hele s results Consider an object with the following parameters: a a 0 = km, i 0 = rad, e 0 = 0 a ω E = rad/s, μ = km 3 s -2, H = 40 km, ρ 0 = kg/m 3, BC = 10 2 kg/m 2 ε ~ 10-8 s -1 The integral can be taken analytically by a Taylor series expansion in n(t) with respect to ε = 0 August 7, rd US-China Tech. Interchange 18

19 Implementation Applying King-Hele s results The Jacobian determinant is For the parameters previously given, a N 1 /N 2 ~ 10-3, N 3 /N 2 ~ 10 7 [s] X/ X 0 1 a Thus, the system can be simplified as a Hamiltonian. a Note that these results are easily transformed into non-singular Poincaré elements August 7, rd US-China Tech. Interchange 19

20 Implementation Results (I) August 7, rd US-China Tech. Interchange 20

21 Implementation Results (I) Calm Stormy 3-σ curve converges upon the distribution of MC points as the order of the STT is increased Uncertainties of stochastic drag force are sufficiently expressed through parametric errors August 7, rd US-China Tech. Interchange 21

22 Implementation Simulation setup (II) Example II Compare results of mean and covariance propagation after days using: a Monte Carlo Compute the mean and variance-covariance matrices based on numerical Monte Carlo results Analytic Approximate as a Hamiltonian system (i.e. energy is conserved) Analytic solution expanded up to 4th order No sampling required August 7, rd US-China Tech. Interchange 22

23 Implementation Results (II) Relative Error (Truth = MC) Scenario a M σ 2 a μ am σ 2 M Calm % % % % % Stormy % % % % % Overall, the analytically propagated cumulants are within 1% relative error for all coordinate directions Covariance can be thought of as a direct measure of the volume of the uncertainty More strongly influenced by the phase volume contraction than the mean August 7, rd US-China Tech. Interchange 23

24 Implementation Results (II) Total Runtime (averaged over 10 runs) Monte Carlo Analytic s s Analytic propagation many orders of magnitude faster than the Monte Carlo Does not scale with propagation time nor sampling resolution of the state space if STT order fixed Analytic techniques potentially both accurate and efficient in non-linearly propagating uncertainty August 7, rd US-China Tech. Interchange 24

25 Conclusions An analytical method of non-linear uncertainty propagation including non-conservative effects was discussed A special solution to the Fokker-Planck equations for deterministic systems State transition tensor concept Applied King-Hele s theory on orbital motion under atmospheric drag Compared with a numerical Monte Carlo simulation with realistic parameter model 8/7/2013 AMOS

26 Conclusions Future work Further incorporate additional perturbations relevant to Earth-orbiting objects J 2 added with semi-empirical method (Parks, 1983) Better (semi-)analytical theories exist 8/7/2013 AMOS

27 Questions? April 2nd, 2013 Thesis Defense (Fujimoto) 27