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Analytic Non-Gaussian Confidence Boundary Method for Chance-Constrained Trajectory Control

Ethan Burnett, Spencer Boone

Abstract

Standard chance constrained control algorithms typically rely on the assumption that uncertainties in vehicle states obey Gaussian statistics. Highly nonlinear systems tend to disrupt Gaussianity, challenging standard chance-constrained control methods. This paper develops a non-Gaussian confidence boundary parameterization technique for such cases where the problem departs appreciably from the Gaussian assumption. The approach is to consider the true confidence boundary as a perturbation of the one predicted from covariance, deriving perturbed boundary geometry from computed higher-order statistical moments. Applying this technique to so-called "banana-shaped distributions" (found e.g. in orbital mechanics problems) enables a simple parameterization of the confidence boundary using the skew and kurtosis tensors. The method is then applied to an impulsive stochastic spacecraft maneuver targeting problem in two-body dynamics. An algorithmic implementation outperforms a standard linear covariance-based approach in computing control parameters satisfying certain probabilistic bounds on the non-Gaussian distribution.

Analytic Non-Gaussian Confidence Boundary Method for Chance-Constrained Trajectory Control

Abstract

Standard chance constrained control algorithms typically rely on the assumption that uncertainties in vehicle states obey Gaussian statistics. Highly nonlinear systems tend to disrupt Gaussianity, challenging standard chance-constrained control methods. This paper develops a non-Gaussian confidence boundary parameterization technique for such cases where the problem departs appreciably from the Gaussian assumption. The approach is to consider the true confidence boundary as a perturbation of the one predicted from covariance, deriving perturbed boundary geometry from computed higher-order statistical moments. Applying this technique to so-called "banana-shaped distributions" (found e.g. in orbital mechanics problems) enables a simple parameterization of the confidence boundary using the skew and kurtosis tensors. The method is then applied to an impulsive stochastic spacecraft maneuver targeting problem in two-body dynamics. An algorithmic implementation outperforms a standard linear covariance-based approach in computing control parameters satisfying certain probabilistic bounds on the non-Gaussian distribution.

Paper Structure

This paper contains 16 sections, 2 theorems, 25 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Model
  4. Chance Constrained Control
  5. Approximating Non-Gaussian Confidence Bounds
  6. Gaussian confidence contours
  7. Moment tensors and Gaussian identities
  8. Modeling weakly non-Gaussian contours
  9. First correction: bend (banana centerline curvature)
  10. Second correction: long-axis asymmetry
  11. Final Result
  12. Numerical Example
  13. Dynamics model: two-body problem
  14. Scenario
  15. Optimization with chance constraints
  16. ...and 1 more sections

Key Result

Proposition 1

Under Assumption ass:bend_ansatz, the least-squares coefficients minimizing $\mathbb{E}\!\left[(\hat{v}-\beta-\alpha \hat{u}^2)^2\right]$ satisfy $\beta=-\alpha$ and In the Gaussian case, $\alpha=\beta=0$, by consequence of Remark rem:gaussian_identities. These coefficients provide the best fit of the assumed distribution for the specified ansatz. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure C1: Non-Gaussian chance constraints
  • Figure D1: Asteroid maneuver targeting scenario
  • Figure D2: Monte Carlo results for linear covariance ellipse chance constraints
  • Figure D3: Monte Carlo results for banana contour chance constraints

Theorems & Definitions (13)

  • Definition 1: State chance constraint
  • Definition 2: Gaussian confidence contour
  • Remark 1: Standard parameterization
  • Remark 2: Symmetries of moment tensors
  • Remark 3: Gaussian moment identities
  • Remark 4: Banana distributions
  • Proposition 1: Quadratic bending coefficients
  • proof
  • Remark 5: Quartile Asymmetries
  • Definition 3: Cornish-Fisher Expansion
  • ...and 3 more