Darin Koblick, Raytheon
Keywords: Two-body Analytic Solution, Closed Form, Min/Max Altitude, Extrema
Abstract:
Community interest in satellites, missiles, and hypersonic vehicles traveling through low altitude regions has grown in recent years, especially for missions harnessing atmospheric drag to station keep or change orbital planes. Mission designers are often required to determine the vehicle min/max altitude (extrema) to avoid risky maneuvers which dip too far into a planets atmosphere, hit tall mountain ranges, impact the planet surface, or traverse environments with high levels of radiation.
Over the last half century, many methods were developed to search and compute close encounter distances between satellites, orbits, and celestial bodies. Some may use analytic solutions to determine close approach distances of comet trajectories to outer planets, minimum distances between two Keplerian orbits, and minimum orbit intersection distances; others compute the minimum approach distances between multiple bodies numerically (e.g., probability of collision or conjunction). However, no previously developed methods were capable of directly computing the altitude extremum for a trajectory.
In this research, a novel closed-form analytical solution is proposed for computing the surface altitude extremum of any two-body orbit. This novel approach is extended to accommodate trajectories over oblate spheroids using Halley’s method (a root finding technique with cubic convergence) on a functional representation of altitude with respect to true anomaly. Several important applications in astrodynamics that require extremal altitude constraints include asteroid impact deflection (a kinetic impactor strikes the surface normal to a desired velocity change), low altitude flyby missions (primary objective often includes high resolution imagery of surface features), and lambert transfers (trajectories that intersect a central body are considered infeasible).
For demonstration purposes two example trajectory applications are described and solved using hand calculations following the process outlined in our closed-form procedure section. The first example application contained a slightly eccentric low Earth orbit (e.g., a low flying Planet Labs CubeSat), and the second consisted of a hyperbolic trajectory (e.g., Cassini Earth gravity assist flyby). For each application, the altitude extremal values were compared with their corresponding function derivatives and the solution accuracy was verified for both spherical and oblate spheroid cases.
Our proposed closed form solution was tested on thousands of orbits consisting of various eccentricities (circular through hyperbolic), inclinations (retrograde through prograde), and segment lengths (several degrees through a full orbital period) using both spherical Earth and World Geodetic System 1984 (WGS-84) ellipsoid models to compute surface altitudes. Runtime performance comparisons were made against the MATLAB implementation of fminbnd, a golden section search root finding algorithm to determine the altitude extremal values. A comparison of results revealed that our proposed closed-form analytical approach accelerated computational runtime performance by three to five orders of magnitudewhile maintaining accuracies within several centimeters of numerically computed altitude extrema.
Improving the runtime performance of flight algorithms that solve trajectory optimization and control problems provides tangible advantages for future missions as it allows additional fuel and time savings while improving solution accuracy. This novel method is an ideal candidate for flight hardware implementations as it requires little computational overhead, few lines of code, and a small memory footprint.
Date of Conference: September 14-17, 2021
Track: Astrodynamics