Polynomial Constraint Station-Keeping

Source: Guo Jianyu (2020) "Research on libration point orbit design and station-keeping strategies based on the two-dominant invariant manifold method"

Website: https://cislunarspace.cn

Definition

Polynomial Constraint Station-Keeping is an orbit maintenance technique based on the two-dominant invariant manifold method. This method establishes nonlinear polynomial relationships among the three-directional motions of a periodic orbit near a libration point and uses these relationships as constraints for orbit correction and control, without requiring a pre-designed nominal orbit.

Core Principles

Comparison with Traditional Methods

Orbit maintenance methods are generally divided into two categories:

Method TypeDescriptionCharacteristics
"Tight" controlDesign a target orbit; the spacecraft must return to within a certain range of the target orbit after deviationHigh control accuracy, but depends on pre-designed nominal orbit
"Loose" controlNo target orbit is specified; the spacecraft only needs to remain near the libration pointHigher computational cost, but more flexible

Polynomial Constraint Method

The innovation of the polynomial constraint station-keeping method lies in:

  1. No pre-designed nominal orbit required: Corrections are based on the current orbital state of the spacecraft
  2. Real-time capability: Uses real-time measured orbital states compared against polynomial constraints
  3. Flexibility: Allows deviations within a certain range, better adapting to complex dynamical environments

Mathematical Formulation

Let the three-directional states of a periodic orbit be (x,y,z)(x, y, z); then the polynomial constraint relationship is:

z=f(x,y)=∑i,jcijxiyjz = f(x, y) = \sum_{i,j} c_{ij} x^i y^j

When the actual orbital state deviates from the polynomial constraint, control corrections are applied to bring the orbit back to the constraint surface.

Application Scenarios

The polynomial constraint station-keeping method is applicable to:

  • Halo orbit maintenance: Long-term maintenance of Halo orbits at Earth-Moon L1/L2 points
  • Lissajous orbit maintenance: Orbit control of quasi-periodic orbits
  • Complex dynamical environments: Accounting for solar gravitational perturbations, lunar disturbances, and other influencing factors

Simulation Verification

Guo Jianyu (2020) conducted simulation verification using the following models:

  • Earth-Moon system nondimensional model
  • Sun-Earth system dimensional model
  • Dimensional model with lunar disturbances

Simulation results demonstrate that the polynomial constraint method exhibits good performance in both orbit maintenance accuracy and computational efficiency.

Core Elements

Mathematical Definition

Polynomial constraint station-keeping establishes nonlinear polynomial relationships among three-directional periodic orbit motions through the two-dominant invariant manifold method, using these as constraint conditions for real-time orbit correction without requiring a pre-designed nominal orbit.

Key Properties

Compared to traditional "tight" and "loose" control methods, the polynomial constraint method offers better real-time capability and flexibility, better adapting to station-keeping requirements in complex dynamical environments.

Numerical Methods

The polynomial relationships are used to compute target state values, and thruster corrections are applied based on measured state deviations, with typical correction periods ranging from several days to several weeks.

Application Value

The polynomial constraint station-keeping method provides a new orbit control strategy for libration point missions with the following advantages:

  • Reduced computational complexity: No need to solve complex trajectory optimization problems
  • Improved real-time performance: Rapid correction based on current state
  • Enhanced robustness: Better adaptability to model errors and measurement deviations

References

  • Guo J Y. Research on libration point orbit design and station-keeping strategies based on the two-dominant invariant manifold method[D]. Beijing University of Technology, 2020. (in Chinese)
  • Breakwell J V, et al. A "broken-rails" steering law for stationkeeping of libration point orbits[R]. 1974.
  • Simo C, Gomez G. Station keeping of a quasi-periodic halo orbit using invariant manifolds[C]. 1986.