New Approach to Halo Orbit Determination and Control

A simple, iterative numerical method for the determination of halo orbits of the circular restricted three-body problem is developed. The nonlinearities inherent to the halo orbit problem are treated as trajectory-dependent, persistent disturbance inputs. The method then utilizes a disturbance accommodating, linear state-feedback controller for the computation of a trajectory about a libration point that can be used as a fuel-efficient nominal path. It is also shown that the method can be used as an iterative method for generating a large, complex, quasiperiodic Lissajous trajectory starting with a first-order reference trajectory. N the restricted three-body problem, originally formulated by Euler in 1772, two massive bodies, called primaries, exhibit twobody motion about their center of mass, called the barycenter. A third body with negligible mass is introduced into this system and does not affect the motion of the two primaries; its motion is determined by the gravitational field of the primaries. The equilibrium points, called libration or Lagrangian points, are points where the gravitational and centrifugal forces acting on the third body cancel each other. Euler had shown the existence of three collinear libration points in 1765 and Lagrange discovered the two triangular libration points in 1772. The existence of periodic halo orbits around such libration points has been known for many years and has been the focus of much research in celestial mechanics. In the late 1960s and early 1970s, the use of libration points in the Earth-moon-spacecraft system was extensively studied by Farquhar1'2 and Farquhar and Kamel.3 Because the same face of the moon always faces the Earth, communications with the far side of the moon is impossible without a relay network. One method to provide this communications network, introduced by Farquhar, would be to position one communications satellite in a halo orbit about the translunar L2 point. In addition, if another communications satellite is located at the cislunar L\ point, there could be continuous communications coverage between the Earth and most of the lunar surface. Halo orbits have also received attention for actual space missions. In November 1978 a spacecraft called the International Sun-Earth Explorer-3 was placed into a halo orbit around the interior sunEarth LI point.4'5 It remained in this orbit until June 1982. One of its mission objectives was to continuously monitor the characteristics of the solar wind and other solar induced phenomena, such as solar flares, about an hour before they could disturb the space environment near the Earth. Much of the research dealing with halo orbits centered on describing the reference orbital trajectory accurately. This is of particular importance in reducing the fuel expenditure for a spacecraft forced to follow the reference orbital path.1"6 The reference orbits often take the form of a series solution to the nonlinear equations of motion. In this paper, a new approach to the orbit determination and control problem of a spacecraft in a halo orbit is presented. If a spacecraft is forced to follow a nominal path derived with the linearized equations of motion, it is subject to the dynamical nonlinearities