Libration point stationkeeping using the θ-D technique

Abstract

Unstable orbits about the collinear libration points require stationkeeping maneuvers to maintain the nominal path. A new method for stationkeeping such unstable orbits is proposed here using continuous thrust. The stationkeeping challenge is formulated as a nonlinear optimal control problem in the framework of the circular restricted three-body problem with the Sun and Earth/Moon center of mass as the two primaries. A recently developed control technique known as the “θ-D controller” is employed to provide a closed-form suboptimal feedback solution to this nonlinear control problem. In this approach an approximate solution to the Hamiltonian-Jacobi-Bellman (HJB) equation is found by adding perturbation terms to the cost function. The controller is designed such that the actual (numerically integrated) trajectory tracks a predetermined Lissajous reference orbit with good accuracy. Numerical results employing this method demonstrate the potential of this approach with stationkeeping costs varying between 0.52 ∼ 0.68 m/s per year (the range depending on the particular simulation parameters used), which is of the same order of magnitude as other methods using discrete maneuvers with halo orbits. The costs are modest and the method provides flexibility in selecting the “tightness” of the control versus fuel consumption. The algorithm is well-suited for integration with onboard flight software, as the nonlinear optimal control law is solved in closed-form and the Riccati equation must only be solved once, resulting in a computationally efficient controller.