Kustaanheimo–Stiefel Variables for Planetary Protection Compliance Analysis

Abstract

Planetary protection in trajectory design aims to assess the impact probability of space mission disposed objects based on their initial uncertain conditions, to avoid contaminating other planetary environments. High precision dynamical models and propagation methods are required to reach high confidence levels on small estimated impact probabilities. These requirements have so far confined planetary protection analyses to robust Monte Carlo-based approaches, using the Cartesian formulation of the full force problem dynamics. This work presents the improvements brought by adopting the Kustaanheimo-Stiefel (KS) formulation of the dynamics. The KS formulation is combined with reference frame switch procedures and adaptive non-dimensionalization upon detection of close encounters. The fibration property of the KS space, namely the parametrizable locus of point arising when mapping to a higherdimensional space, is exploited to minimize the computational time, because of the minimized numerical stiffness of the system. Impact probability estimation tasks become more efficient than the single simulation case, since the regularization of the dynamics removes the singularity of gravitational potentials for distances approaching zero. Despite an almost halved computational burden for Monte-Carlo analysis, the precision of the single simulations is increased by nearly one order of magnitude, setting a new performance benchmark for planetary protection tasks.

GPU-based Augmented Trajectory Propagation: orbital regularization interface and NVIDIA CUDA Tensor Core performance

“…Other than the reduction of the steps taken presented in [6], the reason for this additional performance enhancement is explained by the lower amount of states that CUDAjectory's output handlers need to manage, resulting in an overall reduced memory and output processing overhead. In fact, the runtime reduction without trajectory storage remains consistently around the expected 40 %.…”

“…However, a more complex dynamics that does not necessarily follow a dominantly two-body trajectory could not benefit from the KS regularization if it was kept in its standard form. Moreover, the barycentric formulation of the dynamics builds a more efficient simulation already in the Cartesian form, since tidal terms are not present [6]. With the purpose of building a simulation setup that remains as general as possible in the cases it can efficiently tackle although still featuring the core benefits of the KS formulation, a barycentric formulation of the KS equations of motion is derived.…”

“…with i p identifying the primary body, included in the definition of the two-body energy ϵ, r i and µ i position vector with respect to the primary and gravitational parameter of the i-th body, respectively. A few modifications are required to write the dynamics centered in the barycenter of the N bodies involved [6]. In general, through the regularization, smaller physical time steps are implicitly taken in the proximity of the center of the reference frame.…”

GPU-based high-precision orbital propagation of large sets of initial conditions through Picard–Chebyshev augmentation

“…This accuracy requirement has so far limited the analysis to Monte-Carlo based techniques [5][6][7][8][9] with Cartesian equations of motion, whose simulations inherently carry a heavy computational burden that may limit the mission development time. Recent developments [10] nearly halved the required computational time by implementing the Kustaanheimo-Stiefel [11] formulation of the dynamical motion. This work takes inspiration from the results obtained in [12,13], whose aim was to optimize a multi-flyby trajectory arc while taking advantage of the natural dynamics of the N-body relativistic environment.…”

“…The modified PC method has been continuously developed in the past few years, both in its formulation and implementation and outlining possible applications for Earth orbits where it contributed to increase the efficiency of the numerical analyses. Junkins et al [14] analyzed the performances of the method comparing the efficiency against the Runge-Kutta-Nystrom 12 (10) integrator, proposing also a second order version. Later, Koblick and Shankar [15] extended the analysis to the propagation of accurate orbits testing difference force models with NASA's Java Astrodynamics toolkit.…”