Probability of Collision Estimation and Optimization Under Uncertainty Utilizing Separated Representations


Many current applications of maneuver design to astrodynamics consider a deterministic case, where statistics or uncertainty is left unquantified. When including constraints based on the probability of collision, any solution must be robust to the uncertainty of the system. This paper considers the methodology of separated representations for orbit uncertainty propagation and its subsequent application to a reliability design formulation of the maneuver design problem. Separated representations is a polynomial surrogate method that has been shown to be both efficient at propagating uncertainty when considering high stochastic dimension and accurate over long propagation times. This efficiency is leveraged to improve tractability when solving the reliability design problem using optimization under uncertainty. Two sequential, potential collisions are considered in the results of this paper, with one object able to maneuver. The optimization problem therefore seeks to avoid both collisions. The probability of each collision is estimated via large numbers of samples propagated via the separated representation. The accuracy of the surrogates is compared to that of a Monte Carlo reference, and the variability of the estimated probabilities of collision is analyzed.

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  1. 1.

    For this particular application, the setting change for the minimum difference used to calculate finite-difference gradients is the DiffMinChange variable within fmincon’s optimoptions function


α :

Index indicating the considered object, i.e., 1, 2, or 3

α CI :

Significance level of the confidence interval

β :

Index for the collision in considering, i.e., 1 or 2

B :

Cumulative Beta distribution function

c :

Polynomial coefficients for the separated representation

χ :

The reliability index that represents the lower bound of an optimization under uncertainty constraint

\(\bar {\chi }\) :

The reliability index resulting from optimization under uncertainty

CP :

Clopper-Pearson confidence interval bounds

d :

Total input dimension of the test case and surrogate

\(\mathcal {D}\) :

Training data set for the separated representation

ΔV :

Maneuver velocity vector

ΔV 0 :

Nominal collision avoidance maneuver

ΔV (β) :

Designed maneuver to satisfy optimization under uncertainty conditions and avoid collision(s) β

d 𝜃 :

Dimension of all of the design input directions

d ξ :

Dimension of random input directions

η :

The random inputs associated with the state of object 1

P :

Maximum order of orthogonal polynomials

p :

Indexing for orthogonal polynomial order

\({P}_{c}^{(\beta )}\) :

Time integrated probability of collision for collision(s) β (optional)

ψ :

Polynomials orthogonal to the distributions of separated representation inputs

\(\boldsymbol {q}^{(\beta )}_{\alpha }\) :

Full state vector of the quantities of interest for object α at collision β

\(\boldsymbol {q}_{\alpha }^{(\beta )}\) :

Separated Representation of the quantities of interest for object α at collision β

\(\mathcal {R}\) :

Defined keep out radius for collision detection

r :

Rank of the separated representation

r α :

The velocity vector of object α

r α :

The position vector of object α

s :

Separated representation normalization coefficients

s :

The distance between a unique sample pair

Σ α :

State uncertainty covariance matrix for object α

\(\boldsymbol {\Sigma }^{\Delta V}_{\alpha }\) :

The maneuver uncertainty covariance matrix for object α

\(\mathcal {T}\) :

Time of closest approach

t :

The time variable

t 0 :

Test case epoch time

\(\mathcal {T}^{(\beta )}_{j}\) :

Unique time of closest approach for sample pair j at collision β

t ΔV :

The time of the collision avoidance maneuver

𝜃 :

A vector of design inputs for the optimizer and surrogate

𝜃 (β) :

Design inputs that satisfy the optimization under uncertainty formulation while avoiding collision(s) β (optional)

t k :

Discrete time steps used to aid the Brent optimizer in searching for \(\mathcal {T}\)

u :

Univariate functions in the separated representation

u 0 :

Vector of deterministic factors for the separated representation

Ξ :

Deterministic inputs to which the design inputs 𝜃 are mapped

ξ :

A vector of random input directions. The result of combining η and ω

y :

The complete vector of input directions for the test case as well as the surrogate


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The material for the work by Marc Balducci is provided by the NSTRF fellowship, NASA Grant NNX15AP41H. The authors would like to thank Dr. Ryan Russell for his optimization algorithm advice, as well as Dr. Alireza Doostan for his knowledge with regards to surrogate convergence.

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Balducci, M., Jones, B.A. Probability of Collision Estimation and Optimization Under Uncertainty Utilizing Separated Representations. J Astronaut Sci (2020).

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  • Conjunction assessment
  • Separated representations
  • Reliability design
  • Optimization under uncertainty