Mioara Joldes

Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

This article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature.

On the computation of the reciprocal of floating point expansions using an adapted Newton-Raphson iteration

Many numerical problems require a higher computing precision than that offered by common floating point (FP) formats. One common way of extending the precision is to represent numbers in a multiple component format. With so-called floating point expansions, numbers are represented as the unevaluated sum of standard machine precision FP numbers. This format offers the simplicity of using directly available and highly optimized FP operations and is used by multiple-precisions libraries such as Baileys QD or the analogue Graphics Processing Units tuned version, GQD. In this article we present a new algorithm for computing the reciprocal FP expansion a-1 of a FP expansion a. Our algorithm is based on an adapted Newton-Raphson iteration where we use “truncated” operations (additions, multiplications) involving FP expansions. The thorough error analysis given shows that our algorithm allows for computations of very accurate quotients. Precisely, after q ≤ 0 iterations, the computed FP expansion x = x0 + ... + x2q-1 satisfies the relative error bound |x-a-1/a-1|≤2-2q(p-3)-1, where p > 2 is the precision of the FP representation used (p = 24 for single precision and p = 53 for double precision).

Rigorous uniform approximation of D-finite functions using Chebyshev expansions

A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-known that the order-n truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions. We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.

Probabilistic Collision Avoidance for Long-term Space Encounters via Risk Selection

This paper deals with collision avoidance between two space objects involved in a long-term encounter, assuming Keplerian linearized dynamics. The primary object is an active spacecraft - able to perform propulsive maneuvers - originally set on a reference orbit. The secondary object - typically an orbital debris - is passive and represents a threat to the primary. The collision avoidance problem addressed in this paper aims at computing a fuel-optimal, finite sequence of impulsive maneuvers performed by the active spacecraft such that instantaneous collision probability remains below a given threshold over the encounter and that the primary object goes back to its reference trajectory at the end of the mission. Two successive relaxations are used to turn the original hard chance-constrained problem into a deterministic version that can be solved using mixed-integer linear programming solvers. An additional contribution is to propose a new algorithm to compute probabilities for 3-D Gaussian random variables to lie in Euclidean balls, enabling us to numerically validate the computed maneuvers by efficiently evaluating the resulting instantaneous collision probabilities.

A New Method to Compute the Probability of Collision for Short-term Space Encounters

This article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter. In this specific framework of conjunction, classical assumptions reduce the probability of collision to the integral of a 2-D normal distribution over a disk shifted from the peak of the corresponding Gaussian function. Both integrand and domain of integration directly depend on the nature of the short-term encounter. Thus the inputs are the combined sphere radius, the mean relative position in the encounter plane at reference time as well as the relative position covariance matrix representing the uncertainties. The method presented here is based on an analytical expression for the integral. It has the form of a convergent power series whose coefficients verify a linear recurrence. It is derived using Laplace transform and properties of D-finite functions. The new method has been intensively tested on a series of test-cases and compares favorably to other existing works.