APPLICATION OF UDWADIA-KALABA FORMULATION TO THE PLANAR ELLIPTICAL RESTRICTED THREE-BODY PROBLEM

This paper introduces the Udwadia-Kalaba (UK) formulation of constrained dynamics as applied to the planar elliptical restricted three-body problem (Sun-Earth-spacecraft). The unconstrained motion of the spacecraft is analyzed under the gravitational influences of the Sun and the Earth, where the Earth is in its elliptical orbit around the Sun. In addition, solar radiation pressure is applied to the spacecraft. The results verify a drift from the Lagrange point. Then, the UK formulation is applied to derive the equation of motion of the spacecraft by adding the constraints such that the spacecraft remains exactly at the Lagrange point with respect to the Sun and the Earth. Exact amount of thrust with the directions is obtained to stay at the Lagrange points L1 and L4. The Baumgarte’s stabilization method is included to consider the case when the spacecraft is inserted away from the Lagrange point initially. In this case, the results are analyzed for the underdamped, overdamped and critically-damped cases. In addition, the V is compared for the transient response with the time-varying linear quadratic regulator (LQR) where the optimal feedback control gains are designed via the periodic Riccati equation. The control accelerations required to maintain the desired position obtained by the UK technique are obtained to be identical to those obtained via using the feedforward part of the time-varying LQR control, as expected.