Yashica Khatri, University of Colorado Boulder; Daniel Scheeres, University of Colorado Boulder
Keywords: Conjunction Analysis, Gaussian Mixture Model, State Transition Tensor, Simplified Dynamical System, Monte Carlo, Space Situational Awareness, Uncertainty Propagation
Abstract:
This work presents a conjunction toolkit applicable to short-term and long-term conjunctions that can be accessed using a git repository and takes inputs such as orbital parameters and generates a probability of collision using a semi-analytical uncertainty propagation conjunction analysis (SAUPCA) toolkit.
Due to an increase in the number of objects in space, current sensors are overloaded with the need of recurrent observations. Restrained observation needs mean that the objects can be observed in a limited capacity. To statistically estimate the future location of an object in case of limited observations, the state and uncertainty need to be propagated from the epoch to determine conjunction possibilities. Previously, Monte Carlo methods have been primarily used to propagate this information forward and perform conjunction analysis between such objects. Although accurate, these methods come with a high associated computation burden because of the extensive numerical propagation need. This makes research into fast and accurate semi-analytical tools that combine accurate uncertainty propagation techniques with relevant dynamics essential.
A Gaussian distribution associated with the sensor measurement outputs loses its Gaussian nature when propagated over time due to the high nonlinearity associated with orbit dynamics. In the SAUPCA toolkit, the initial uncertainty distribution is split into a multidimensional Gaussian Mixture Model (GMM) to achieve smaller distributions that maintain linearity over longer propagation times. These individual components are propagated forward using State Transition Tensors (STTs) to accurately capture the loss of Gaussianity. These STTs are computed using numerical propagation performed through STT Equations of Motion (EOMs) which are computed with a Simplified Dynamical System (SDS). The SDS uses Hamiltonian equations to separate mean dynamics and short-period variation dynamics. The mean dynamics allow fast propagation and the short-period variation addition at final times capture the oscillatory components associated with the propagation performed with the full dynamics. This toolkit currently has the options to utilize Keplerian dynamics in addition to J2 and SRP effects. The methodology and components of this method have been shown in previous works by Khatri and Scheeres.
The conjunction assessment is performed using a simple double integral over a circle in the encounter plane. This method is valid for short-term conjunctions. It is also valid for long-term conjunctions when combined with the GMM-STT uncertainty propagation model, which is shown in previous works by the authors. The probability of collision is evaluated between individual components of the GMM and combined using a double-weighted sum based on the GMM splitting. This conjugate probability of collision can be used as actionable information to determine collision risks.
This combined uncertainty propagation and conjunction analysis is presented and displayed in a concise SAUPCA toolkit that can take a range of orbital elements for the two objects in conjunction and output a probability of collision. This probability of collision matches the Monte Carlo probability of collision to within a relative error percentage. Examples are shown using a variety of orbital element to showcase the validity of the method and the wide-range of inputs that can be used with the toolkit. The presentation of this work in a toolkit user-accessible form is novel and has a wide-variety of applications in the SSA domain.
Date of Conference: September 19-22, 2023
Track: Conjunction/RPO