Sam Wishnek, University of Colorado Boulder; Marcus Holzinger, University of Colorado Boulder; Patrick Handley, Ball Aerospace
Keywords: IOD, Initial Orbit Determination, Prescribed Orbits, Particle Swarm
Abstract:
A novel, empirically-validated method for robust initial orbit determination from two or more observations of an object’s angles and angle rates is presented. The method can be applied for all orbital regimes including those where traditional methods, such as Gauss, double-r, and Laplace fail. It also works for all times-of-flight to produce the corresponding set of feasible orbit solutions using admissible regions.
Methods such as Gauss’s rely on approximations that break down for longer duration times-of-flight. Other methods such as double-r are highly sensitive to the initial range guesses. Most methods only accept angle observations as inputs and require three observations to estimate a state. However, angle rates can be pulled from optical measurements in the direction and magnitude of the imaged streaks; neglecting this information results in coarser orbit solutions and requires 3 measurements, rather than 2. The proposed method uses full two body dynamics, retains the benefits of prescribed orbits in its reduced initial guess sensitivity compared to double-r, and accepts angles and angle rates to build a state estimate with only two observations.
The proposed method starts as an adaption of prescribed orbits (R. Karimia, D. Mortari, 2011) re-derived to accept angle rates and angles instead of only angles. The prescribed orbits method uses a least squares optimization to find a solution that satisfies an equality based on the universal variable formulation of two body dynamics. Like the original prescribed orbits, the method works well for scenarios where the observer is located inside of the target’s orbit, yet fails for geostationary observers observing other geostationary objects as well as objects orbiting closer to the Earth. The solution space contours of a cost function based on this approach show that the solution space has several local minima that can divert the algorithm from the correct solution for geometries where the target orbit lies inside the observer orbit.
The new proposed cost function has a unique non-trivial zero at the correct orbit solution. Because the system is overdetermined with the minimum two measurements, the global minimum will not approach zero if the two observations do not correspond to the same orbiting object. Applying a particle swarm descent method to the problem allows the system to converge to the correct state for any orbital regime. In its most natural implementation this problem is a four-dimensional search. However, by projecting one measurement into the space of the other, it can be solved with a pair of two-dimensional searches which is much more computationally efficient and parallelizable. Applying admissible regions to both observations allows the state space to be further constrained with a corresponding improvement to the computational efficiency. In addition, the particle swarm approach can report all of the feasible states in the multi-revolution case. The proposed method overcomes the shortcomings of many traditional initial orbit determination methods and has been empirically verified to find solutions to cases well-approximated by two-body dynamics.
Furthermore, this work includes a comparison of the proposed and existing methods for initial orbit determination to highlight the domains in which the various methods excel. While the particle swarm approach of the proposed method sacrifices computation time in comparison to the single iteration of the previously established methods, it consistently converges to the correct solution or set of possible solutions independent of the orbital regime.
Date of Conference: September 15-18, 2020
Best Student Paper Award Winner 2020
Track: Astrodynamics