Fuel-Optimal Geometric Path Planning Algorithm for Spacecraft Formation Flying

A novel geometric guidance algorithm is proposed for spacecraft formation flying. The controllability of the full relative motion is analyzed using Lie bracket theory. The fuel-optimal two-point boundary value problem is reframed into a curve minimization on a Riemannian manifold. The optimal path is shown to be a geodesic of the solution manifold, and it is obtained by solving a partial differential equation. The large convergence radius of the proposed method is robust to bad initial guesses. The geometric guidance is used in a multiple-spacecraft architecture, and a collision avoidance scenario illustrates how the geometric path planner solves constrained path planning problems. Finally, the proposed guidance is compared to a regulator path planner to showcase the low fuel requirement associated with traveling along the geodesic.