Siwei Fan, Purdue University; Alex Friedman, Purdue University; Carolin Frueh, Purdue University;
Keywords: Light Curves, Shape Recovery, Inversion
Abstract:
Obtaining information beyond the orbit and the center of mass of space objects is critical for the solution of a number of challenges in Space Situational Awareness (SSA). This includes reliable propagation in the presence of non-conservative forces, determining operational status, the detection and resolution of failures and to judge capabilities and intend of satellites. One integral part of characterization information is shape. Shape information is present in so-called light curve measurements, time-series of brightness measurements of the same object. The problem is that it is superposed with attitude, materials and depends upon the observation geometry. The observation geometry can, in many cases, be determined for short time scales largely independent of the characterization information based on astrometric observations and short propagation times, over which the object-specific characteristics effect on the orbit may be neglected. But even when focusing on the shape inversion problem alone, the problem is severely underdetermined and has only a unique solution in the absence of measurement errors for uniform, fully convex shapes, as it was mathematically proven already at the beginning of the 20th century by Russel et al.
This paper has two parts. The first part is focusing on the shape inversion for fully convex objects, the second part expands upon the methodology established in the first part to objects with concavities. Objects with concavities are the most common human-made space objects, for example in the classical box-wing configuration.
In this paper, we show the method and the conditions on the light curve and the object itself of finding an approximation to this theoretically existing unique solution in the presence of realistic measurement noise. This is done via the use of the Extended Gaussian Image EGI), followed by an iterative solution to Minkowski problem. The EGI finds albedo-areas associated with all normal directions on a mathematical sphere. This, however, does not solve the shape inversion, as the adjacency information and the distance of each area of a reference point of the object is not given. This can be determined by the use of Minkowski-Brunn theorem, which states conditions for the Lebegue measure of the mixture of two convex polytopes. In an iterative way, a solution for the support function can be found, allowing to reconstruct the actual shape. This is not sufficient in the presence of measurement noise. Hence, the method is extended via a Multi-Hypothesis approach. Measurement noise is dependent upon the signal-to-noise ratio and hence different in every measurement of within the light curve. The conditions of being able to find the solution under those conditions are determined via the use of the observability Gramian in a new formulation that includes measurement noise and hence closes the gap to the classical information measure. Convergence to the actual solution in the presence of noise without a priori defined shapes in the Multi-Hypothesis approach is shown in examples.
In a final step, the methodology is shown in its extension to non-unique cases, in which the object is not fully convex, but exhibits concavities. There, not even theoretically, a unique solution exists anymore. In our methodology, the convex solution based on the use of EGI and subsequent Minkowski-Brunn theorem is only used as a first guess in an iterative scheme of volume maximization (minimal volume is the compact convex shape). The iteration is done along the characteristic curves. Conditions for convergence to the correct shape with concavities are shown. The performance of the method for realistic satellites shapes with concavities are shown.
Date of Conference: September 17-20, 2019
Track: Non-Resolved Object Characterization