Quadratically Constrained Linear-Quadratic Regulator Approach for Finite-Thrust Orbital Rendezvous

Abstract

This paper focuses on the design of a quadratically constrained linear-quadratic regulator for finite-thrust orbital rendezvous. The original linear-quadratic optimal control problem is subject to maximum thrust magnitude and quadratic collision avoidance constraints. Thrust arcs are approximated by impulsive velocity increments and the Yamanaka–Ankersen transition matrix propagates the state vector. An explicit closed-loop solution is obtained by performing high-order series expansions of the Hamilton–Jacobi–Bellman equation on subregions of the state space associated with specific sets of active constraints. The algorithm is computationally efficient because the Lagrange multipliers are expressed as polynomial functions of the states and can be computed offline. A rendezvous in an elliptical orbit is considered to demonstrate the application of this method.