Utkarsh Mishra, Texas A&M University; Suman Chakravorty, Texas A&M University; Weston Faber, L3Harris; Islam Hussein, Trusted Space, Inc.; Benjamin Sunderland, Kayhan Space; Siamak Hesar, Kayhan Space Corp
Keywords: Initial Orbit Determination, IOD, PAR, CAR, Admissible Region, PGM, Bayesian Update, Information gain, entropy
Abstract:
Initial Orbit Determination (IOD) is a fundamental problem in space exploration and satellite operations that involves determining the initial state of a Resident Space Object (RSO). A single short-arc measurement from a space surveillance sensor like a radar or a telescope provides only partial-state information and is insufficient to initialize all the state parameters of a Resident Space Object (RSO). Traditional schemes for angles-only IOD for Earth-orbiting RSOs include Gauss’s method, Double-r iteration, and Gooding. These methods work with three ordered pairs of topocentric right ascension, and declination measurement [(αt1,δt1),(αt2,δt2),(αt3,δt3)] and their corresponding time stamps ti.
The admissible region is defined as the set of physically acceptable orbits (i.e., orbits with negative energies). With some additional constraints on the orbital semi-major axis, eccentricity, etc, the admissible region can be further constrained, resulting in the Constrained Admissible Region (CAR). If the hard constraints are replaced with a probabilistic representation of the admissible region based on known statistics of the measurement process, it results in the Probabilistic Admissible Region (PAR). PAR+ makes improvements to the original PAR by considering the uncertainty of the constraints to be epistemic and modeling it by possibility functions, to come up with a possibilistic uncertainty representation. More recently it has shown that PAR is equivalent to a bayesian measurement update on the prior pdf of the state of the RSO. For angles-only observation cases, PAR and CAR work with a single right ascension and declination measurement pair (αt1,δt1) and apriori knowledge (constraints or probabilistic description) of some of the orbital elements. Traditional IOD schemes like Gauss’s method as well as admissible region-based methods like CAR and PAR can be used for orbit initiation in Bayesian tracking.
The assumptions that go into each of these orbit initiation schemes like the number of measurements required, the need for apriori knowledge about some orbital elements for admissible region-based techniques, acceptable time separation of measurements in the case of traditional IOD schemes, the treatment of uncertainty in the measurements, etc. vary. The scope of the output of the method i.e., returning a single orbit or a probability density function (pdf) over the possible orbits varies too. One major tradeoff between traditional deterministic IOD and admissible region-based schemes is the need for multiple observations that definitely came from the same RSO (which is complicated in the case of tracking multiple closely spaced objects) versus the need for some apriori knowledge about the state of the RSO that generated the measurement.
This work will present the first detailed comparison between traditional IOD schemes, CAR, and PAR. For the traditional deterministic IOD schemes, in order to get the initial pdf of the state, a Monte Carlo (MC) analysis will be used to map the uncertainty cloud in the measurement space to the full orbital space. In addition to just the three pairs of right ascension and declination measurements discussed earlier, this MC analysis will also use uncertainties in the measurements. The subsequent measurements (after measurement # 3 for deterministic traditional IOD, and after measurement # 1 for CAR and PAR) will be used to recursively update the estimated state using a modern recursive Bayesian filter called Particle Gaussian Mixture (PGM) Filter. The tracking results for different initiation schemes will be compared and contrasted on the structure of the prior pdf (important for re-discovering the object and aiding sensor tasking), the structure of the posterior pdf, and entropy (to quantify the differential in information gain), etc. especially after processing the first three measurements. Results from these empirical studies shed light on the merits of these techniques relative to each other.
The goal of recursive filtering is good tracking and obtaining a more compact pdf of the RSO. A poor choice of initiation scheme can lead to bad initiation of the pdf of the RSO which may lead to the loss of track or difficulties in finding the target because the pdf becomes very diffused. This work will demonstrate that on top of tackling the target-measurement assignment problem, PAR+PGM extracts the most useful information out of the measurements which will be shown by the low entropy after processing the same number of measurements.
Date of Conference: September 19-22, 2023
Track: Astrodynamics