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The halo orbits around the Earth-Moon

For the spatial circular restricted three-body problem (CR3BP), where the two large bodies move in planar circular orbits about their center of mass and a third body of negligible mass moves under only the

There has been a lot of interest in finding and controlling the periodic orbits near the collinear libration points. Farquhar originally proposed using the orbits near the

The techniques for station-keeping of halo orbits can be classified as (1) using continuous control, or (2) generating impulses at discrete time intervals. Breakwell et al. studied station-keeping for translunar communication station using a continuous feedback control to minimize a cost function, which is a weighted combination of position deviation from the nominal orbit and the control acceleration based on a linearization about the reference orbit [

A newly developed numerical computation approach, Modified Chebyshev-Picard Iteration (MCPI) method, is used in this paper for station-keeping to maintain a halo orbit in the Earth-Moon system. This orbit provides an excellent parking orbit for a lunar communication satellite. Details about MCPI methods can be found in the papers by Bai and Junkins [

This paper is structured as follows. We first present the numerical approach to generate the reference halo orbit. We use the third order analytical formulations derived by Richardson [

As is well known, a good nominal reference orbit reduces the fuel cost for unnecessary maneuvers, so the dynamical model used to generate the reference orbit ideally should include the gravitational perturbation forces from the Sun and other planets as well as solar radiation and other forces. Farquhar and Kamel presented fairly detailed equations of motion where the effects of the Sun’s gravitation force, solar pressure, and the Moon’s orbital eccentricity were considered [

Most of the development of the equations of motion can be found in the book by Schaub and Junkins [

Parameters used.

Parameter | Value |
---|---|

System Diagram.

A third-order closed-form solution for the equations of motion is given in Richardson’s paper [

Approximate initial conditions.

Variable | Nondimensional value |
---|---|

−0.390895010335809 | |

0.353556629315019 | |

1.554577497503360 | |

3.336429964438981 |

A nice feature about MCPI methods is that the achieved solutions are represented to high precision in the form of orthogonal Chebyshev polynomials, thus if the coefficients of the states are saved, the state values at arbitrary time can be obtained straightforward (by numerically computing the orthogonal polynomials and then a simple inner product matrix multiplication). We integrate the reference orbit using MCPI method for one orbit and save the coefficients of the position and velocity for future station-keeping reference. An important parameter to choose here is the order of the polynomials to use. Generally, higher order approximation leads to higher accuracy solutions but requires larger storage space to save all the coefficients. We use closure error (after one period) of the reference orbit, defined in (

Closure error versus polynomial order.

Nondimensional

Dimensional

Reference Orbit.

3D

We use MCPI method for station-keeping of the halo orbit generated in the last section. The matrix-vector form of MCPI methods for solving this special type of two-point BVPs, where the positions at the two boundaries are constrained, is shown in Figure

Vector-matrix form of MCPI method for solving second order bvps.

For the current problem, at the time when a maneuver is possible, the current position

The position and velocity of the satellite when it is inserted to the halo orbit will be away from the ideal initial conditions of the reference orbit. The measurement of the position and velocity from Earth will not be perfect. And the implementation of the impulses will also have “execution errors”. We include these three kinds of errors in the simulation. All the errors are assumed to be Gaussian distribution with a mean of zero. The standard deviations of these errors are chosen to be consistent with those used by Gómez et al. [

tracking errors:

insertion errors:

maneuver errors:

We look at station-keeping of the halo orbit for 26 revolutions, which corresponds to 377 days, a little more than one year. 100 Monte Carlo simulations are run and the results in Table

Baseline case results.

Variable | Averaged Results |
---|---|

Max | 27.894 cm/s |

Min | 1.178 cm/s |

Total | 1606.5 cm/s |

Mean deviation distance from the reference | 3.98 km |

Max deviation distance from the reference | 17.59 km |

Maneuver numbers | 182 |

Iterations of MCPI | 3.05 |

Considering most of the thrusters used to generate impulses will have some limit on the minimum

Variable | Averaged results |
---|---|

Max | 29.692 cm/s |

Min | 0 cm/s |

Total | 1523.5 cm/s: about 1476.5 cm/s per year |

Mean deviation distance from the reference | 4.45 km |

Max deviation distance from the reference | 17.38 km |

Maneuver numbers | 168 |

Iterations of MCPI | 3.07 |

Variable | Averaged results |
---|---|

Max | 33.86 cm/s |

Min | 0 cm/s |

Total | 1574.1 cm/s |

Mean deviation distance from the reference | 5.27 km |

Max deviation distance from the reference | 21.29 km |

Maneuver numbers | 155 |

Iterations of MCPI | 3.12 |

Results from One Monte Carlo Simulation.

Impulse history

Deviation

Iteration numbers

Effects of the spacing of maneuvers (averaged from 100 Monte Carlo simulation).

Deviations.

Number of maneuvers

Total impulses

Max impulses

In this section, we study how to reduce the fuel cost using the presented method while maintaining its advantages such as no requirement for any gradient information, computational efficiency, and the final results in an orthogonal polynomial form.

The above-method results in exact satisfaction of terminal boundary conditions on each subinterval if there are no control errors and measurement errors. As a result, the position errors are much smaller than those in the literature [

Performance versus coefficient of control.

Averaged total impulses

Averaged maximum deviation

Averaged mean deviation

Another approach to reduce the fuel cost is to adopt a hybrid propulsion system which uses low thrust during the flight and impulses at some specified points. The hybrid propulsion concept has been studied by Bai et al. for an Earth to Apophis mission design [

Modified Chebyshev-Picard Iteration method is used for station-keeping of a halo orbit around the

Assume the second order dynamic system is described as

This work is supported by the Air Force Office of Scientific Research (Contract number FA9550-11-1-0279).

_{2}Libration point satellite with

_{∞}approach