H2 OPTIMAL HALO ORBIT GUIDANCE

An output feedback guidance law for a halo orbit about the translunar equilibrium point in the circular restricted three-body problem is developed. The equations of motion are derived and the location and stability of the translunar equilibrium point discussed. The halo orbit guidance problem is formulated in the frequency domain from which an output feedback guidance law is developed using HI control theory. Simulation results validate the guidance law and provide data that quantify the effect of control inputs, noise characteristics, and halo orbit characteristics on the steady-state halo orbit stationkeeping costs. ALO orbits were primarily investigated in the late 1960s when NASA began examining follow-on lunar explo- ration opportunities to the Apollo program. In an effort to open up the far side of the moon to exploration, studies were undertaken to determine if continuous communications or staging operations with a far side lunar base could be accom- plished. More recently, a NASA report1 investigated four ap- proaches to a manned mission to Mars. Common to all four approaches was the deployment of a telecommunications satel- lite in translunar halo orbit to support far side lunar communi- cations. Figure 1 shows the geometry of the translunar halo orbit. The translunar equilibrium point is inherently unstable and chaotic; active guidance will be required to maintain the halo orbit. Many investigators have examined the general halo orbit guidance problem; this paper will summarize only those sources that directly apply to the translunar halo orbit guid- ance problem. Farquhar2'3 and Farquhar and Kamel4 provided an extensive review of previous work done on controlling an orbit about an equilibrium point. They considered controlling a halo orbit about the translunar equilibrium point and showed that using a simple proportional plus derivative con- troller provided asymptotic stability while minimizing the con- trol acceleration required. Deviations from the desired orbit were not considered in the minimization. Breakwell et al. 5 formulated the halo orbit guidance problem as a periodic sys- tem. They used the classical optimal control approach with the addition of an observer to the system model. Position devia- tions were considered in their problem formulation, but only results for a large halo orbit radius were given. Subsequently, a flight dynamics study6 of both the halo and hummingbird concepts was completed; the hummingbird con- cept places a spacecraft at a stationary offset position from the equilibrium point. Both concepts were found to be feasible, but the halo orbit was preferred because it has fewer propul- sion requirements for stationkeeping. This study's problem formulation used a frequency matching guidance law with discrete impulses applied twice an orbit. Heppenheimer7 used phase-plane methods to construct a family of locally fuel-opti- mal out-of-plane period controls. Vonbun8 also investigated using a hummingbird orbit rather than a halo orbit and found in general that it required 10% more acceleration to maintain the desired position. More recently, Fraietta and Bond9 com- puted stationkeeping costs for halo orbits about both the cislu- nar and translunar equilibrium points. Finally, Farquhar10 compared the use of a polar lunar orbit and a halo orbit for lunar exploration staging operations. He concluded that a halo orbit space station could offer important operational and performance advantages compared with a po- lar lunar orbit station. Among these advantages were increased communication opportunities with the lunar surface and in- creased launch windows for transfers between the space sta- tion and the lunar surface. This research takes a different approach by using H2 control theory to formulate the halo orbit guidance problem as a con- tinuous thrust system in the frequency domain. The advantage to the frequency domain problem formulation is that it can be extended to modern control theories, such as H2 and H^ the- ory, so that the class of plant disturbances and measurement noises considered can be expanded. This paper presents a guid- ance law that stabilizes the translunar halo orbit and minimizes the position deviation from the halo orbit plus the control acceleration. The relevant equations of motion are given in the next section. Subsequently, the system models, guidance law computation, and simulation results are presented. Finally, the last section summarizes this research and draws general con- clusions.