Utkarsh R. Mishra, Texas A&M University; Suman Chakravorty, Texas A&M University; Islam I. Hussein, Trusted Space; Weston Faber, L3Harris Technologies; Siamak Hesar, Kayhan Space Corporation; Benjamin Sunderland, Kayhan Space Corporation
Keywords: PAR, CAR, MOT, MTT, Multi Target Tracking, GMM, IOD
Abstract:
A single measurement from an SSA sensor like a radar or a telescope gives only partial-state information. In an ideal single object tracking scenario, Initial Orbit Determination (IOD) schemes piece together multiple observations over time from the same object and try to fit a state vector. This method assumes that the series of observations came from the same Resident Space Object (RSO). But in the Multiple Object Tracking (MOT) scenarios, with multiple observations in each frame, there would be combinatorial growth in the possible ‘hard’ observation-to-observation associations. It is also considerably easier to mathematically evaluate the likelihood of an observation associated with a cataloged object compared to keeping track and evaluating the likelihood of associating each measurement (among many) in one frame to each measurement (among many) in a second frame. In fact for the overwhelming majority of multiple space-object tracking scenarios, a mathematically rigorous formulation for weighing one observation-to-observation versus another is impossible. This is because most objects appear physically the same to the sensor and even if identifying features like reflectivity and radar cross-sections could be measured making associations using them remains ad-hoc.
The admissible region is the set of physically acceptable orbits that can be constrained even further if additional constraints on some orbital parameters like semi-major axis, eccentricity, etc, are present. This results in the constrained admissible region (CAR). If hard constraints are replaced, based on known statistics of the measurement process, with a probabilistic representation of the admissible region, it results in the probabilistic admissible region (PAR). PAR can be used for orbit initiation in Bayesian tracking.
In the angles only case, the PAR algorithm tries to map the guessed uncertainties in a few orbital parameters (a, e, i, Ω) and measurements (α, δ) into the probability density function (pdf) of the states. This pdf can then be used as the initial distribution of the states of the object and get propagated and recursively updated using a Bayesian filter. This pdf turns out to be multimodal and can only be handled by filters that can handle multimodal pdf representation like the Particle Gaussian Mixture (PGM) filter.
The original algorithm to solve PAR involves mapping the uncertainty in a few orbital parameters (a, e, i, Ω) and measurements right ascension and declination (α, δ) into a particle cloud in range (ρ), range-rate (dρ/dt), and angle rates (dα/dt, dδ/dt), where ρ is the range of the space object from the sensor. This involved solving a complicated root solving problem for the angular momentum vector and then verifying the resulting energy and eccentricity values. This paper shows that in fact there is a very simple closed-form solution that can be obtained by geometry and fundamental astrodynamics. First draw samples (a, e, i, Ω, α, δ)j from the distribution on (a, e, i, Ω) and (α, δ). Then the idea is to use the line of sight given by a sample of measured angles (α, δ)j, and the orbital plane defined by a sampled value of inclination, and right ascension of ascending node ( i, Ω)j to find the distance from the center of the Earth (i.e. the focus) to the RSO. Finally, the sampled value of a and e i.e., (a, e)j can be used to give the shape of the ellipse. It is shown that only two possible orientations of the ellipse on the orbital plane can satisfy the distance from the center of the Earth (i.e. the focus) to the RSO that was found earlier. These are the two particles in state space (x, y, z, dx/dt, dy/dt, dz/dt)1j , and (x, y, z, dx/dt, dy/dt, dz/dt)2j satisfy the sampled values of (a, e, i, Ω, α, δ)j. Repeating this simple geometrical mapping for a large number of (a, e, i, Ω, α, δ)j gives a particle cloud in the state space. This gives the pdf of the states that happen to be multimodal. This geometric solution to PAR called G-PAR is a big improvement over the original PAR in ease of implementation, giving clear reasoning for the resulting multimodal pdf. This paper also resolves the question of why some (a, e, i, Ω, α, δ)j fail to produce any corresponding particles in state space due to incompatibility among the sampled values.
This paper presents and discusses the results of G-PAR – based initialization for angles-only observation of objects in a variety of orbital regimes. The paper concludes with a brief discussion on how G-PAR can be integrated with a PGM filter to recursively update the pdf of the RSO initialized with G-PAR.
Date of Conference: September 27-20, 2022
Track: Astrodynamics