Laura Pirovano, The University of Auckland, Te P?naha ?tea – Space Institute; Roberto Armellin, The University of Auckland, Te P?naha ?tea – Space Institute
Keywords: Space Traffic Management, Maneuver Detection, Convex Optimization
Abstract:
Collecting and maintaining knowledge of resident space objects involves several tasks, ranging from observation acquisition and short-arcs data association to orbit determination (OD), among a few. Once acquired, the knowledge of a space object needs to be updated following a dedicated observing schedule. However, dynamics mis-modeling and unknown maneuvers can alter the accuracy of the catalog, resulting in uncorrelated tracks originating from the same object. A further step is thus needed to regain correlation and keep an accurate catalog. This paper presents the chain of algorithms we developed to perform initial orbit determination, tentative data association assuming natural motion, orbit determination, and finally maneuver estimation applied to optical observations of MEV-2 during its low-thrust orbit raise. The algorithms use high-order methods, and the maneuver is estimated through convex optimization without a-priori assumption on the thrust arcs structure and thrust direction. This exercise stemmed from the Phantom Echoes 2 (PE2) experiment within a Five-Eyes initiative, for which optical observations of MEV-2 and INTELSAT 10-02 during rendezvous and proximity operations were gathered to improve allied capabilities for the protection of spacecraft in GEO.
Admissible State Region (ASR): To obtain an initial guess for very short arcs (VSAs), a set of optical observations is linearly regressed to compute a four-dimensional attributable. The two remaining degrees of freedom, forming the well-known Admissible Region (AR), are then bounded with energy constraints. We use high-order techniques to analytically describe the uncertainty of the AR and attributable, reaching the concept of ASR, where the 6D uncertainty is described with a list of polynomials [Pirovano2021]. Thanks to this formulation, it is easy to estimate the uncertainty bounds for each domain, which can be exploited for data association. When more than 3 observations (the minimum required to perform linear regression) are available, updating the ASR by calculating the minimum residual between the real and expected observations is possible. Sub-domains for which the residual is too large are then pruned.
Orbit Set (OS): When tracks are longer than VSAs, it is possible to obtain a candidate state guess by fitting a Keplerian orbit through three observations. By choosing the first, middle, and last observation in a track, it is possible to exploit the full length of the track to obtain a candidate solution. Each angular observation, though, has a specific precision. The OS algorithm then finds all possible orbits that geometrically fit the three observations within the specified accuracy [Pirovano2020]. Like the ASR, the output is an analytical expression for which it is easy to estimate uncertainty bounds. When tracks approach the VSAs size, the projected OS’s size tends to the AR’s size. Conversely, when more observations are available, the updated ASR tends to the size of the OS.
Range intersection: For both ASRs and OSs, the uncertainty bounds are easy to retrieve thanks to their polynomial form. It is then possible to compare the uncertainty ranges in classical orbital elements for different guesses to assess correlation. Under the assumption that the elements’ variation within the time window is much smaller than the uncertainty involved, a 5D intersection is sought on the slowly varying elements without propagation. If the intersection is the empty set, then the observations are not correlated. Otherwise, a reference orbit can be obtained within the intersection and refined through a Least Squares (LS) technique.
Differential Algebra Least Squares (DALS): When two ASRs and/or OSs are correlated, or when the track is longer than a VSA – like the MEV-2 case – it is possible to refine the state guess through a LS routine that uses all observations in the track. In this work, we use the Differential Algebra Least Squares (DALS) [Losacco2021], a high-order LS method that can perform OD, with the option to estimate the Bfactor and SRP coefficients.
Convex Optimization Maneuver Estimation Technique (COMET): If two guesses end up being uncorrelated following the range intersection method, there is no ballistic path that can connect the two states. However, the satellite may have maneuvered in the meantime, meaning an un-modeled perturbation affected the ballistic hypothesis. We devised a new method based on second-order cone programming to find the optimal maneuver profile, if it exists, to connect two states with uncertain bounds – their covariances – effectively solving an uncertain two-points boundary value problem. The 4th order Conjugated Uncertainty Transform (CUT-4) [Adurthi2018] is used to analyze the effect of uncertainty in the states on the overall maneuver estimation.
The algorithms’ chain is used to analyze optical observations obtained by Defense Technology Agency (DTA) within the PE2 experiment, firstly obtaining ASRs and OSs, understanding if maneuvers were performed between observing windows through range intersection, then performing ODs and estimating the actual maneuvers with COMET.
[Adurthi2018] Adurthi, N., Singla, P., and Singh, T. “Conjugate Unscented Transformation: Applications to Estimation and Control.” (2018)
[Losacco2021] Losacco, M., Principe, G., Armellin, R., Pirovano, L., Gondelach, D., San Juan, J.F., Lara, M., Dominguez Gonzalez, R., Pina Caballero, F., and Urdampilleta, I. “Differential Algebra-based Orbit Determination with the Semi-analytical Propagator SADA” (2021)
[Morselli2014] Morselli, A., Armellin, R., Di Lizia, P., Bernelli Zazzera, F., “A high order method for orbital conjunctions analysis: Sensitivity to initial uncertainties” (2014)
[Pirovano2020] Pirovano, L., Santeramo, A.D., Armellin, R., Di Lizia, P., and Wittig, A. Probabilistic data association: the orbit set (2020).
[Pirovano2021] Pirovano, L., Armellin, R., Siminski, J., Flohrer, T. “Differential algebra enabled multi-target tracking for too-short arcs” (2021)
[Wittig2014] Wittig, A., Di Lizia, P., Armellin, R., Bernelli Zazzera, F., Makino, K., and Berz, M. “An automatic domain splitting technique to propagate uncertainties in highly nonlinear orbital dynamics” (2014)
Date of Conference: September 27-20, 2022
Track: Astrodynamics