Brandon A. Jones

Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos

This paper demonstrates the use of polynomial chaos expansions for the nonlinear, non-Gaussian propagation of orbit state uncertainty. Using linear expansions in tensor products of univariate orthogonal polynomial bases, polynomial chaos expansions approximate the stochastic solution of the ordinary differential equation describing the propagated orbit, and include information on covariance, higher moments, and the spatial density of possible solutions. Results presented in this paper use non-intrusive, i.e., sampling-based, methods in combination with either least-squares regression or pseudospectral collocation to estimate the polynomial chaos expansion coefficients at any future point in time. Such methods allow for the usage of existing orbit propagators. Samples based on sun-synchronous and Molniya orbit scenarios are propagated for up to ten days using two-body and higher-fidelity force models. Tests demonstrate that the presented methods require the propagation of orders of magnitude fewer samples ...

Comparisons of the Cubed-Sphere Gravity Model with the Spherical Harmonics

The cubed-sphere gravitational model is a modification of a base model, e.g., the spherical harmonic model, to allow for the fast evaluation of acceleration. The model consists of concentric spheres, each mapped to the surface of a cube and combined with an appropriate interpolation scheme. The paper presents a brief description of the cubed-sphere model and a comparison of it with the spherical harmonic model. The model was configured to achieve a desired accuracy so that dynamical tests, e.g., evaluation of the integration constant, closely approximate that of the spherical harmonic model. The new model closely approximates the spherical harmonic model, with propagated orbits deviating by a fraction of a millimeter at or above feasible Earth-centered altitudes.

Postmaneuver Collision Probability Estimation Using Sparse Polynomial Chaos Expansions

This paper describes the use of polynomial chaos expansions to approximate the probability of a collision between two satellites after at least one performs a translation maneuver. Polynomial chaos provides a computationally efficient means to generate an approximate solution to a stochastic differential equation without introducing any assumptions on the a posteriori distribution. The stochastic solution then allows for orbit state uncertainty propagation. For the maneuvering spacecraft in the presented scenarios, the polynomial chaos expansion is sparse, allowing for the use of compressive sampling methods to improve solution tractability. This paper first demonstrates the use of these techniques for possible intraformation collisions for the Magnetospheric Multi-Scale mission. The techniques are then applied to a potential collision with debris in low Earth orbit. Results demonstrate that these polynomial chaos-based methods provide a Monte Carlo-like estimate of the collision probability, including ad...

Orbit Propagation Using Gauss-Legendre Collocation

This paper discusses the application of Gauss-Legendre collocation for orbit propagation. Previous presentations of collocation propagators for astrodynamics used a xed-step implementation, thereby limiting their eectiveness for eccentric orbits. The presented formulation uses a variable-step implementation, thereby improving its use for scenarios where the necessary time step changes over time. Additionally, a combination of low- and high-delity force models reduce the overall computation time for the integration. The method was compared to other commonly employed integration tools for both two-body and higher-delity propagation. Several design parameters in the variable-step implementation limit the robustness of the method, but comparisons with the common explicit techniques imply some gains may be made with future development. These tests only considered a serial implementation of the Gauss-Legendre propagator, but parallel methods and their implications for computationally ecient orbit propagation are also discussed.

Measurement-based Birth Model for a Space Object Cardinalized Probability Hypothesis Density Filter

The cardinalized probability hypothesis density (CPHD) multi-target filter allows for the state estimation of known and previously unknown objects in challenging observation environments. To identify new objects, the CPHD requires a birth model to define the density of potential new targets in the single-target space. This paper presents an observation-based birth model using the constrained admissible region that identifies previously unknown space objects and instantiates a Gaussian mixture representation of the new target density. The Gaussian mixture formulation of the CPHD then refines the state estimate of the new target while simultaneously tracking known targets. Two variations of a simulation test demonstrate the efficacy of the method for the tracking of objects in the near-geosynchronous region using groundand space-based optical sensors. The first test case shows the filter’s performance for dense clutter and a priori knowledge of the new target’s presence. The second variation demonstrates the filter’s ability to track known and unknown objects with no a priori knowledge of the new target and a reduced clutter density. This latter case demonstrates that the CPHD with the new birth model correctly identifies potential new targets, discards false detections, and maintains custody of the previously known objects in the multi-target state.

Variation of delivered impulse as a function of asteroid shape

This paper will review recent research results focused on the effect of realistic asteroid shapes on the linear momentum delivered to an asteroid during a mitigation attempt. We use simple models for the effect of kinetic impactors and convolve these with a realistic asteroid shape model. For the asteroid shape we use a radar-derived shape model for the asteroid Golevka that captures global topography. For a given impact site we use realistic error distributions and determine how variable the delivered linear and angular momentum impulse is. We find strongly non-Gaussian deviations in delivered momentum, indicating that it may be difficult to achieve a desired level of precision in a deflection attempt.

The CPHD Filter With Target Spawning

In its classical form, the cardinalized probability hypothesis density (CPHD) filter does not model the appearance of new targets through spawning, yet there are applications for which spawning models more appropriately account for newborn objects when compared to spontaneous birth models. In this paper, we propose a principled derivation of the CPHD filter prediction step including spontaneous birth and spawning. A Gaussian Mixture implementation of the CPHD filter with spawning is then presented, illustrated with three applicable spawning models on a simulated scenario involving two parent targets spawning a total of five objects.

Sequential Orbit Determination with the Cubed-Sphere Gravity Model

The cubed-sphere model provides rapid evaluation of the gravity field for more efficient orbit propagation. This paper characterizes the improved computational efficiency of sequential orbit determination, specifically the extended and unscented Kalman filters, when using this new model instead of the common spherical harmonic model. To use the new gravity model with the extended Kalman filter, capabilities to represent the Jacobian of the gravity acceleration are added to numerically integrate the state transition matrix. Filter tests consider improvements for several simulated satellite scenarios with several combinations of measurements provided for estimation. Since cubed-sphere models of higher degree require only a slight change in computation time, orbit propagation and determination systems may now use this model to improve fidelity without any significant change in cost. Using the cubed-sphere model reduces the computational burden of the orbit determination process, with larger benefits found for high-degree filter models. Differences in the estimated trajectories when using the disparate gravity models remain several orders of magnitude less than the absolute filter error for the cases examined.

Concept for a new frontiers mission to Ganymede: A Planetary Science Summer School study

As part of the NASA Planetary Science Summer School 2010, the Ganymede Interior, Surface and Magnetosphere Observer (GISMO) team developed a robotic mission to Ganymede, one of Jupiters icy moons. This process included the formulation of the science objectives and the selection of a payload tailored to meet these goals. The team then designed a mission architecture aimed toward achieving the science objectives. Using a sequence of 14 flybys of Ganymede, the vehicle would use a simple, staged operation of the science payload. This timeline allows for a simplified design, with relatively low risk and cost. Principle challenges included the finite power available to the vehicle, along with a limited data downlink rate. Otherwise, this preliminary design would meet all mission requirements, as determined by the science goals, and within the allocated cost cap.

Large constellation tracking using a labeled multi-Bernoulli filter

Multiple companies have recently proposed or begun work on large constellations of hundreds to thousands of satellites in low-Earth orbits for the purpose of providing worldwide internet access. The sudden infusion of so many satellites in an already highly-populated orbital regime presents an operational risk to all LEO objects. To enable risk analyses and ensure safe operations, a robust system will be needed to efficiently observe these constellations, and use the resulting data to accurately and precisely track all objects. This paper proposes a rudimentary tasking-tracking system for this purpose. The scheduler uses an information theoretic reward function to determine which high-value tasks, and uses a ranked assignment algorithm to optimally allocate these tasks to a sensor network. The tracking portion employs a labeled multi-Bernoulli filter to process the generated data and estimate the multitarget state of the entire constellation. The effectiveness of this system is demonstrated using a simulated large constellation of 4,425 satellites and a network of six ground-based radar sensors.