Moriba Jah

Probabilistic Initial Orbit Determination Using Gaussian Mixture Models

The most complete description of the state of a system at any time is given by knowledge of the probability density function, which describes the locus of possible states conditioned on any available measurement information. When employing optical data, the concept of the admissible region provides a physics-based region of the range/range-rate space that produces Earth-bound orbit solutions. This work develops a method that employs a probabilistic interpretation of the admissible region and approximates the admissible region by a Gaussian mixture to formulate an initial orbit determination solution. The Gaussian mixture representation of the probability density function is then forecast and updated with subsequent data to iteratively refine the region of uncertainty. Simulation results are presented using synthetic data over a range of orbits, in which it is shown that the new method is consistently able to initialize a probabilistic orbit solution and provide iterative refinement via follow-on tracking.

Initial Orbit Determination using Short-Arc Angle and Angle Rate Data

The population of space objects (SOs) is tracked with sparse resources and thus tracking data are only collected on these objects for a relatively small fraction of their orbit revolution (i.e., a short arc). This contributes to commonly mistagged or uncorrelated SOs and their associated trajectory uncertainties (covariances) to be less physically meaningful. The case of simply updating a catalogued SO is not treated here, but rather, the problem of reducing a set of collected short-arc data on an arbitrary deep space object without a priori information, and from the observations alone, determining its orbit to an acceptable level of accuracy. Fundamentally, this is a problem of data association and track correlation. The work presented here takes the concept of admissible regions and attributable vectors along with a multiple hypothesis filtering approach to determine how well these SO orbits can be recovered for short-arc data in near realtime and autonomously. While the methods presented here are explored with synthetic data, the basis for the simulations resides in actual data that has yet to be reduced, but whose characteristics are replicated as well as possible to yield results that can be expected using actual data.

Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems

Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational b...

An Approach for Nonlinear Uncertainty Propagation: Application to Orbital Mechanics

An approach for nonlinear propagation of orbit uncertainties is discussed while making use of the Fokker-Planck-Kolmogorov Equation (FPKE). The central idea is to replace evolution of initial conditions for a dynamical system with the evolution of a probability density function (pdf) for state variables. The transition pdf corresponding to dynamical system state vector is approximated by using a nite Gaussian mixture model. The mean and covariance of dierent components of the Gaussian mixture model are propagated through the use of an Unscented Kalman Filter (UKF). Furthermore, the unknown amplitudes corresponding to dierent components of the Gaussian mixture model are found by minimizing the FPKE error over the entire volume of interest. This leads to a convex quadratic minimization problem guaranteed to have a unique solution. The two-body problem model with non-conservative atmospheric drag forces and initial uncertainty will be used to show the ecacy of the ideas developed in this paper.

Attitude Determination from Light Curves

T HE unscented Kalman filter (UKF) [1,2] has been used extensively in spacecraft attitude determination based on standard in situ measurements using gyro-based models for attitude propagation [3,4] and, more recently, in orbit determination from groundbased angles and range measurements [5]. In the latter application, the filter is unable to track changes in attitude, because these changes affect the nonconservative forces, which are subtle as compared with the dominant gravitational forces. (i.e., becoming manifest in positional data over long time scales). Attitude changes must be modeled and tracked to properly account for the nongravitational forces experienced by the object. In thisNote,we describe for thefirst time the use of the UKF in attitude determination from ground-based brightness measurements (i.e., light curves), which are very sensitive to attitude variations. This novel application of the UKF allows for the possibility of more accurate and precise orbit determination by exploiting multi-data-type fusion, combining both astrometric and photometric observationswithin the samedata-reduction process [5].

Coupled orbit-attitude dynamics of high area-to-mass ratio (HAMR) objects: influence of solar radiation pressure, Earth’s shadow and the visibility in light curves

The orbital and attitude dynamics of uncontrolled Earth orbiting objects are perturbed by a variety of sources. In research, emphasis has been put on operational space vehicles. Operational satellites typically have a relatively compact shape, and hence, a low area-to-mass ratio (AMR), and are in most cases actively or passively attitude stabilized. This enables one to treat the orbit and attitude propagation as decoupled problems, and in many cases the attitude dynamics can be neglected completely. The situation is different for space debris objects, which are in an uncontrolled attitude state. Furthermore, the assumption that a steady-state attitude motion can be averaged over data reduction intervals may no longer be valid. Additionally, a subset of the debris objects have significantly high area-to-mass ratio (HAMR) values, resulting in highly perturbed orbits, e.g. by solar radiation pressure, even if a stable AMR value is assumed. Note, this assumption implies a steady-state attitude such that the average cross-sectional area exposed to the sun is close to constant. Time-varying solar radiation pressure accelerations due to attitude variations will result in un-modeled errors in the state propagation. This work investigates the evolution of the coupled attitude and orbit motion of HAMR objects. Standardized pieces of multilayer insulation (MLI) are simulated in a near geosynchronous orbits. It is assumed that the objects are rigid bodies and are in uncontrolled attitude states. The integrated effects of the Earth gravitational field and solar radiation pressure on the attitude motion are investigated. The light curves that represent the observed brightness variations over time in a specific viewing direction are extracted. A sensor model is utilized to generate light curves with visibility constraints and magnitude uncertainties as observed by a standard ground based telescope. The photometric models will be needed when combining photometric and astrometric observations for estimation of orbit and attitude dynamics of non-resolved space objects.

Nonlinear uncertainty propagation for perturbed two-body orbits

The main objective of this paper is to present the development of the computational methodology, based on the Gaussian mixture model, that enables accurate propagation of the probability density function through the mathematical models for orbit propagation. The key idea is to approximate the density function associated with orbit states by a sum of Gaussian kernels. The unscented transformation is used to propagate each Gaussian kernel locally through nonlinear orbit dynamical models. Furthermore, a convex optimization problem is formulated by forcing the Gaussian mixture model approximation to satisfy the Kolmogorov equation at every time instant to solve for the amplitudes of Gaussian kernels. Finally, a Bayesian framework is used on the Gaussian mixture model to assimilate observational data with model forecasts. This methodology effectively decouples a large uncertainty propagation problem into many small problems. A major advantage of the proposed approach is that it does not require the knowledge o...

Space object shape characterization and tracking using light curve and angles data

This paper presents a new method, based on a multiple-model adaptive estimation approach, to determine the most probable shape of a resident space object among a number of candidate shape models while simultaneously recovering the observed resident space object’s inertial orientation and trajectory. Multiple-model adaptive estimation uses a parallel bank of filters, each operating under a different hypothesis to determine an estimate of the physical system under consideration. In this work, the shape model of the resident space object constitutes the hypothesis. Estimates of the likelihood of each hypothesis, given the available measurements, are provided from the multiple-model adaptive estimation approach. The multiple-model adaptive estimation state estimates are determined using a weighted average of the individual filter estimates, whereas the shape estimate is selected as the shape model with the highest likelihood. Each filter employs the unscented estimation approach, reducing passively collected ...

Collision Probability with Gaussian Mixture Orbit Uncertainty

In view of the high value of space assets and a growing space debris population increasingly caused by random collisions in a congested space environment, collision analysis is of great significance. Collisions between space objects (spacecraft and space debris) can at best be determined in a probabilistic manner because of the lack of perfect knowledge about the parameters and motions of the space objects. Collision analysis based on the nominal orbital parameters without taking into account the uncertainty in those parameters or in the orbit motions is inaccurate. The probability of collision between two space objects provides a quantitative measure of the likelihood that the space objects will collide with each other. Collision probability is closely related to close approaches, 3 collision detection, and conflict probability and has been derived for low Earth orbits (including the International Space Station), Assistant Professor, Department of Mechanical and Aerospace Engineering. Email: demarsk@mst.edu. Senior Member AIAA. Assistant Professor, Department of Aerospace Engineering. Email: cheng@ae.msstate.edu. Associate Fellow AIAA. Senior Research Engineer. Associate Fellow AIAA.

Refining Space Object Radiation Pressure Modeling with Bidirectional Reflectance Distribution Functions

High-fidelity orbit propagation requires detailed knowledge of the solar radiation pressure on a space object. The solar radiation pressure depends not only on the space object’s shape and attitude, but also on the absorption and reflectance properties of each surface on the object. These properties are typically modeled in a simplistic fashion, but are here described by a surface bidirectional reflectance distribution function. Several analytic bidirectional reflectance distribution function models exist, and are typically complicated functions of illumination angle and material properties represented by parameters within the model. In general, the resulting calculation of the solar radiation pressure would require a time-consuming numerical integration. This might be impractical if multiple solar radiation pressure calculations are required for a variety of material properties in real time; for example, in a filter where the particular surface parameters are being estimated. This paper develops a method...