Gauss's Variational Equation-Based Dynamics and Control for Formation Flying Spacecraft

Abstract

Formation flying is an enabling technology for many future space missions, and this paper presents several modeling and control extensions that would enhance the efficiency of many of these missions. In particular, a new linear time-varying form of the equations of relative motion is developed from Gauss’s variational equations. These new equations of motion are further extended to account for the effects of J2, and the linearizing assumptions are shown to be consistent with typical formation flying scenarios. It is then shown how these models can be used to initialize general formation configurations and can be embedded in an online, optimization-based, model predictive controller. A convex linear approach for initializing fuel-optimized partially J2 invariant orbits is developed and compared with analytic approaches. All control methods are validated using a commercial numerical propagator. Thesimulationresultsillustratethatformation flyingusingthismodelpredictivecontrollerwithJ2-modifiedGauss’s variational equations requires fuel use that is comparable to using unmodified Gauss’s variational equations in simulations that do not include the J2 effects. Nomenclature a = semimajor axis b = semiminor axis e = eccentricity h = angular momentum i = inclination M = mean motion n = orbit frequency p = semilatus rectum r = magnitude of radius vector � = argument of latitude � = right ascension of the ascending node ! = argument of perigee